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T ��� � � �� � �@��qK T � � �! ����R �c � ��@��@� k �� � �@��{A��� ��_�English_ophelp english_ophelp1.c 0 Performs decimal arithmetic (which is not exact). Example: $\sqrt 2 = 1.414214$ Example: 2^(1/2) = 1.414214 Example: ln 2.0 = 0.69315. Also evaluates sin, tan, etc. Factor an integer (less than 4 billion). Example: $360 = 2^3\times 3^2\times 5$. You will be prompted to enter a value of the variable (or variables) Replace $\pi $ by an approximate decimal value, 3.14159235... Replace $e$ by an approximate decimal value, 2.718281828... Compute a numerical value of a function using the definition of the function. Example: x^3-x+1 = (x+1.32472)(x^2 - 1.32472 x + 0.754878) Evaluate a Bernoulli number to a rational number Evaluate an Euler number to a rational number Change some decimals to fractions. Use with caution on approximate values. Example: 64 = 8^2 Example: 1000 = 10^3 Example: 256 = 4^4. You will be prompted to enter the exponent. Example: 256 = 4^4. You will be prompted to enter the base. Examples: 36 = 6^2, or 256 = 2^8. Example: 3 is selected, you enter 2, the result is 2 + 1. This is the most important property of the complex number i. Examples: i^4 = 1, i^8 = 1, i^12 = 1 Examples: i^5 = i, i^9 = i, i^(-3) = i Example: i^6 = -1 Example: i^7 = -i Perform exact arithmetic (but not exponentiation) on complex numbers. Example, $(1+i)^2 = \sqrt 2 i$. Perform exact arithmetic (including exponentiation) on complex numbers. Perform approximate decimal-number arithmetic involving complex numbers. Factor an integer into Gaussian prime-power factors, e.g. 5 = (1+2i)(1-2i) Example: -3+4i = (1+2i)^2 Example: $\sqrt $i = 0.707168 + 0.707168 i Example, i^(1/2) = 0.707168 + 0.707168 i Example, cos i = 1.543080635 Show the value of an expression after you enter values for the variables. Drop double minus signs. Example: -(x^2 - 2x + 1) becomes x^2 + 2x - 1 Example: -x-5 becomes -(x+5) Use the associative law. Example: (a+b) + (c+d) = a+b+c+d Brings terms of a sum to standard order. Example: y+x = x+y Example: x^2 + 0 + 5 = x^2 + 5 Example: x^2 + x + sin x - x = x^2 + sin x Example: x^2 + 3x + 2x = x^2 + 5x Example: x^2 + 3x + 2x^2 + 2x = 3x^2 + 5x Commutative law: reverse the order of summation in the selected term. Example: 5(1-x) becomes -5(x-1) Example: -5x becomes 5(-x) Example: -5xy becomes 5x(-y) Example: 5x(-y)z becomes 5xy(-z) Example: $2^100\times 0$ becomes 0 Drop factors of 1. Pull minus signs to the front of a product. Use the associative law. Example: (3x^2)(yz) = 3x^2yz Example: $2x\times 3y$ = 6xy Put factors in a product into standard order. Example: yx = xy Use the law x^n x^m = x^(n+m). Example: x^2x^3 = x^5. Distributive law. Example: x(x^2 + 1) = x^3 + x. Example: (x-2)(x+2) = x^2-4 Example: (x+3)^2 = x^2 + 6x + 9 Example: (x-3)^2 = x^2 - 6x + 9 Example: (x-1)(x^2+2x+1) = x^3-1 Example: (x+1)(x^2-2x+1) = x^3+1 Commutative law: reverse the order of terms in a product Example: (x+1)(x+2) = x^2 + 3x + 2 Multiply out products of sums in the numerator, but not in the denominator. Multiply out products of sums in the denominator, but not in the numerator. Example: 3x = x + x + x Zero divided by anything nonzero is zero. Anything divided by 1 is unchanged. Definition of reciprocal. Example, $2 \times (1/2) = 1$ Example, (3/4)(x/y) = 3x/(4y) Example, 3(x/2) = 3x/2 Example: x^2 y / x = xy Add fractions with the same denominator by adding the numerators. Break a fraction whose numerator is a sum into two or more fractions. Break $(a\pm b)/c$ if one of the resulting fractions will cancel. Example: (x^2 + 2x + 2)/(x+1) = x+1 + 1/(x+1) Cancel the greatest common factor of numerator and denominator. Example: 2x/3y = (2/3)(x/y) Example: $(x^2 + y^2)/\sqrt 2 = (1/\sqrt 2) x^2 + y^2$ Example: $3e^(it)/\sqrt 2 = (3/\sqrt 2) e^(it)$ Example: ax/(2y) = (a/2)(x/y) Example: $\sqrt 3x/2 = (\sqrt 3/2)x$ Cancel a minus sign from numerator and denominator. Push a minus sign into the numerator. Push a minus sign into the denominator. Pull a minus sign out of the numerator. Pull a minus sign out of the denominator. Pull minus signs out of a sum in the numerator. Pull minus signs out of a sum in the denominator. Change the order of terms in the denominator and adjust the sign. Example: (1-x)/(3-x) = (x-1)/(x-3) Example: 2x/3 = 2(x/3) Example: 1/(x(1-x^2)) = (1/x)(1/(1-x^2) Example: x/2 /(y/2) = x/y Example: 3/(2/x) = 3x/2 Example: 1/(2/x) = x/2 Example: (3/2)/x = 3/(2x) Example: (2/3)/x = (2/3)(1/x) Example: (2/3)x/y = 2x/3y Example: 1/(x^2+2x+1) = 1/(x+1)^2 Use common denominators on a sum of fractions inside a bigger fraction. Example: 1/x + 1/y = 1/x(y/y) + (1/y)(x/x) Same as find common denom, but ignores non-fractions in a sum. Example: (x/2)(y/3) = xy/6 Example: 2(x/y) = 2x/y Put factors of a product in standard order. Example: yx = xy Example: 1/x + 1/y + 1 = (y+x+xy)/(xy) Example: 1/x + 1/y + 1 = (y+x)/(xy) + 1 Example: y/x + x/y = (x^2+y^2)/xy Ignores non-fractions in the sum, working only on the fractions. You specify what to multiply by. Example, x/y = x^2/xy if you enter x. Anything to the zero power is 1; except 0^0 is undefined. The first power of x is just x. Zero to any positive power is zero. 1 raised to any power is 1. Examples: (-1)^4 = 1 and (-1)^3 = -1 $c\in Z$ means that c is an integer. Here the number $a$ has to be positive. Provided the new numerator and denominator are defined. Example: (2x)^2 = 4x^2 Example: (x+1)^2 = x^2+2x+1 Example: (x+1)^3 = x^3 + 3x^2 + 3x + 1 Example: 3^(2+x) = 3^2 3^x Example: a^2/b^2 = (a/b)^2 Example: x^5/x^3 = x^2 Example: x^3/x^5 = 1/x^2 Example: (x+1)^2 = (x+1)(x+1) Example: (x+1)^3 = (x+1)(x+1)(x+1) Example: (x+1)^4 = (x+1)(x+1)(x+1)(x+1) Example: x^5 = x^2 x^3. You enter the 2 when prompted. Example: (x+1)^2 = x^2 + 2x + 1 Example: (x-1)^3 = x^3 - 3x^2 + 3x - 1 Example: 2^(2n)=(2^2)^n Example: 2^(2n)=(2^n)^2 Example: 2^(2nm) = 2^(2n)^m Example: 1/2^n = (1/2)^n Eliminate a constant negative exponent Eliminate a negative exponent. Eliminate a negative exponent. Example: x^(-2) = 1/x^2 Eliminate a negative exponent. Example: x^(-2)/3 = 1/(3x^2) Eliminate a negative exponent in the denominator. Example: 1/x^(-2) = x^2 Eliminate a negative exponent in the denominator. Example: 3/x^(-2) = 3x^2 Example: 2/x = 2x^(-1) Example: (2/x)^(-2) = (x/2)^2 Example: x^(n-2) = x^n/x^2 Provided both sides are defined. Example: $\sqrt 2\sqrt x = \sqrt (2x)$ Provided both sides are defined. Example: $\sqrt (2x) = \sqrt 2\sqrt x$ Example: $\sqrt (4y) = 2\sqrt y$ Square and square root are inverses, so long as x is nonnegative. If you don't know the sign of x, you need the absolute value sign. Example: $\sqrt 8 = \sqrt 2^3$ Provided both sides are defined. Example: $\sqrt (x/2) = \sqrt x/\sqrt 2$ When the signs of x and y are not known, you need the absolute value sign. Provided both sides are defined. Example $\sqrt x/\sqrt 2 = \sqrt (x/2)$ Since $\sqrt x \sqrt x = x$ by definition of $\sqrt $. Of course, x must be nonnegative. Example, $(\sqrt x)^6 = x^3$ Example, $(\sqrt x)^5 = x^2\sqrt x$ Compute square roots if the value is a rational number. Example, $\sqrt 16 = 4$ Compute approximate decimal values of square roots. Example, $\sqrt 2$ = 1.41416... Does not compute square roots or roots; performs (other) arithmetic. Example: $\sqrt (x^2+2x+1)/\sqrt (x^2-1) = \sqrt (x+1)^2/\sqrt (x-1)(x+1)$ Example: $\sqrt (x^2+2x+1) = \sqrt (x+1)^2$ Example: $1/(1-\sqrt x) = (1+\sqrt x)/((1-\sqrt x)(1+\sqrt x))$ and so later to $(1+\sqrt x)/(1-x)$ Example: $(1-\sqrt x)/(1+\sqrt x) = (1-\sqrt x)(1+\sqrt x)/(1+\sqrt x)^2$ and so later to $(1-x)/(1+\sqrt x)^2$ If you don't know the sign of x, the absolute value sign is necessary. Example: $\sqrt (2x)/\sqrt 2 = \sqrt x$ Multiply out products of sums occurring inside a square root. The operation a^2-b^2 = (a-b)(a+b) will not create a new root; this one will. $^2\sqrt $ and $\sqrt $ are two symbols with the same meaning. Example: $\sqrt x = ^4\sqrt x^2$. You will be prompted to enter n. Example: $\sqrt x = (^4\sqrt x)^2$. You will be prompted to enter n. Example: $\sqrt x^4 = x^2$ Example: $\sqrt x^5 = x^2 \sqrt x$ The factor outside the root must be nonnegative. Example: $1/(1-\sqrt x) = (1+\sqrt x)/(1-x)$ Express a fractional exponent of $\onehalf $ as a square root. Example: $a^(5/2) = \sqrt (a^5)$ Example: $a^(5/3) = ^3\sqrt (a^5)$ Express a square root using an exponent of $\onehalf $ Express a root using a fractional exponent. Example: $^3\sqrt x^2 = x^(2/3)$ Example: $(^3\sqrt x)^2 = x^(2/3)$ Example: $(\sqrt x)^3 = x^(3/2)$ Express $1/\sqrt x$ using a negative fractional exponent. Express the reciprocal of a root using a negative fractional exponent Example: (-1)^(5/3) = -1. Does not use complex roots. Example: 8^(2/3) = (2^3)^(2/3) Example: x/x^(1/3) = (x^3/x)^(1/3) Example: x^(1/3)/x = (x/x^3)^(1/3) Example: x^(n/2) = (\sqrt x)^n Example: x^(n/3) = (^3\root x)^n Example: $^3\sqrt 5^3\sqrt x = ^3\sqrt (5x)$ Example: $^3\sqrt (2x) = ^3\sqrt 2 ^3\sqrt x$ Example: $^3\sqrt x^2 = (^3\sqrt x)^2$ Example $^3\sqrt x^5 = x ^3\sqrt x^2$ Example: $^3\sqrt (x^3) = x$ Example: $^3\sqrt x^6 =x^2$ Example: $^6\sqrt x^3 = \sqrt x$ Example: $^9\sqrt x^3) = ^3\sqrt x$ Example: $(^3\sqrt x)^3 = x$ Example: $(^3\sqrt a)^2 = ^3\sqrt (a^2)$ Example $(^3\sqrt a)^8 = a^2 ^n\sqrt a^2$ Example: $^3\sqrt 12 = ^3\sqrt (2^2\times 3)$ Example: $^3\sqrt (-a) = -^3\sqrt a$, n odd Perform arithmetic, evaluating roots to rational values if possible. Example: $^3\sqrt (x^3+3x^2+3x+1) = ^3\sqrt (x+1)^3$ Multiply out sums of products under a root sign. Example: $\sqrt (\sqrt 2) = ^4\sqrt 2$ Example: $\sqrt (^3\sqrt 2) = ^6\sqrt 2$ Example: $^3\sqrt (\sqrt 2) = ^6\sqrt 2$ Example: $^3\sqrt (^4\sqrt 2) = ^(12)\sqrt 2$ Write a root of a quotient as a quotient of roots Write a quotient of roots as a root of quotients Example: $x/^3\sqrt x = (^3\sqrt x)^2$ Example: $^3\sqrt x/x = 1/(^3\sqrt x)^2$ Example: $^3\sqrt (2x)/^3\sqrt (2y) = ^3\sqrt x/^3\sqrt y$ Example: $^n\sqrt (2a)/^n\sqrt a = ^n\sqrt 2$ Find the greatest common divisor of u and v and factor it out of u and v Example: $x^3\sqrt y = ^3\sqrt (x^3y)$ Example: $x^2(^4\sqrt y) = ^4\sqrt (x^8y)$ Example: $-^3\sqrt 2 = ^3\sqrt (-2)$ Example: $x/^3\sqrt x = ^3\sqrt (x^3/x)$ Example: $^3\sqrt x/x = ^3\sqrt (x/x^3)$ Example: $x^2/\sqrt x = \sqrt (x^4/x)$ Example: $\sqrt x/x^2 = \sqrt (x/x^4)$ Example: $(^6\sqrt x)^2 = ^3\sqrt x$ Example: $(^4\sqrt x)^2 = \sqrt x$ Since i^2 = -1, we have 1/i = -i Since i^2 = -1, we have a/i = -ai Since i^2 = -1, we have a/(bi) = -ai/b By definition, i is $\sqrt (-1)$ Example: $\sqrt (-3) = i\sqrt 3$ Example: $1/i^3 = i$ Example: $(x-i)(x+i) = x^2+1$ Factor a sum of squares using complex factors. This is really just the Pythagorean theorem. This is the definition of absolute value of a complex number. Example: $(3 + 5i)/2 = (3/2) + (5/2)i$ Bring a complex number to the standard form $u+vi$ Example: $\sqrt i = sqrt(1/2) + sqrt(1/2) i$ Example: $\sqrt(-i) = sqrt(1/2) - sqrt(1/2) i$ Example: $\sqrt(3+4i) = sqrt((5+3)/2) + sqrt((5-3)/2) i$ Example: $\sqrt(3-4i) = sqrt((5+3)/2) - sqrt((5-3)/2) i$ Example: 2x^2 + 4x + 2 = 2(x^2 + 2x + 1) Example: x^2 + x + 1/4 = (1/4) (4x^2+ 4x + 1) Example: x^3y^2-x^3 = x^3(y^2-1) Example: x^5 - x^3 = x^3(x^2-1) Example: x^2+2x+1 = (x+1)^2 Example: x^2-2x+1 = (x-1)^2 Example: x^2-1 = (x-1)(x+1) Example: x^2-3x+1 = (x-2)(x-1) Example: $x^2-x-1 = (x-1/2-\sqrt 5/2)(x-1/2+\sqrt 5/2)$ Example: x^8 = (x^4)^2 Example: $a^2b^2 = (ab)^2$ Example: $4x^2 + 6x + 9 = 2^2x^2 + 2\times 3x + 3^2$ Factor an integer (less than 4 billion). Example: $360 = 2^3\times 3^2\times 5$ Introduce a new letter by a definition, to simplify the expression. Replace a defined variable by its original definition throughout the line. When solving equations, constants are treated differently than variables. No new variable will be used. Example: x^12 = (x^4)^3 Example: x^12 = (x^3)^4. You enter the 4 when prompted. Factor a difference of cubes. Example: $x^3-1 = (x-1)(x^2+x+1)$ Factor a sum of cubes. Example: $x^3+1 = (x+1)(x^2-x+1)$ Example: x^5-1 = (x-1)(x^4 + x^3 + x^2 + x + 1) Example: x^4-1 = (x+1)(x^3 - x^2 + x - 1) Example: x^5+1 = (x+1)(x^4 - x^3 + x^2 - x + 1) Example: $x^4+1 =(x^2-\sqrt 2x+1)(x^2+\sqrt 2x+1)$ Example (with p=5, q=3): $x^4+x^2+25=(x^2-3x+5)(x^2+3x+5)$ You do not select a term, but let MathXpert try to find a good substitution. You enter a factor, and MathXpert gets the other factor by polynomial division. Systematically tries all possible linear factors with integer coefficients. Break the sum into two groups and factor out their greatest common divisor. Write it as a polynomial in the selected term. Example: 3=x becomes x=3 Example: -x = -3 becomes x = 3 Example: x-3 = 2 becomes x = 5 Example: x+3 = 5 becomes x = 2 Example: x-3 = 5 becomes x = 8 Example: x^2 = x-1 becomes x^2-x+1 = 0 Example: x/2 = x + 1 becomes x = 2x + 2 Example: 2x = 4 becomes x = 2 Example: $\sqrt x = 3$ becomes x = 9 Example: x+y = 3+y becomes x = 3 Example: 2x^2 = 2 becomes x^2 = 1 Example: 3x = 3x becomes 'true' Example: $\sqrt x = -\sqrt x$ becomes x = -x Example: $\sqrt x = -\sqrt x$ becomes $\sqrt x = 0$ Example: $-\sqrt x = \sqrt x$ becomes $\sqrt x = 0$ if ab=0 then a=0 or b=0 quadratic formula $x = -b/2a \pm \sqrt (b^2-4ac)/2a$ complete the square take square root of both sides cross multiply b^2-4ac < 0 => no real roots Use this when the sign of $a$ cannot be determined. Enter a value of the unknown and see the values of the two sides. You will be asked for two values known to bracket a root. Example: x/3 = (x-1)/4 becomes 4x = 3(x-1) Raise both sides to a power. The new equation can have extra roots. Example: x^2 = 9 becomes [x = 3, x = -3] Example: x^3 = 8 becomes x = 2 You will be asked what function to apply to both sides. Put sums involving fractions over a common denominator. Example: (x^2-1)(x-2) = 0 becomes [x^2-1=0, x=2] Example: ax^2=ax becomes [a=0, x^2=x] The other equations will be hidden while you work on the selected one. The equations you hid some time ago will be shown again. Duplicate solutions can be combined. It will work if the proposed substitution eliminates an old variable. Replace a variable by its original definition throughout the line. Example: $x = \sqrt -3$ when seeking real solutions. Some operations may have introduced extra roots which won't check. Example: 3x-1 = x+1 becomes x=1 This substitution will eliminate the quadratic term. The discriminant of a cubic equation cx^3+ax+b is $D = b^2/4c + a^3/27c^3$. Repeats the cubic equation so you can continue working on it. This substitution will make the equation quadratic in y^3. in cx^3+ax+b=0: $x=^3\sqrt (-b/2c+\sqrt D)+^3\sqrt (-b/2c-\sqrt D)$ where D = b^2/4c + a^3/27c^3. in cx^3-ax+b=0: $x=[2\sqrt (a/3)cos(t/3),2\sqrt (a/3)cos(t+2pi/3),2\sqrt (a/3)cos(t+4pi/3)]$ where $cos t = -b/(2c)\sqrt (27/a^3)$. in cx^3+ax+b=0: $x=[^3\sqrt (-b/2c+\sqrt D)+^3\sqrt (-b/2c-\sqrt D),(1/2)^3\sqrt (-b/2c+\sqrt D)+^3\sqrt (-b/2c-\sqrt D) \pm (\sqrt 3/2)(^3\sqrt (-b/2c+\sqrt D)-^3\sqrt (-b/2c-\sqrt D)]$ Make a substitution $x = f(u)$ where $x$ is an old variable and $u$ is new. Eliminate a defined variable using its definition. Example, change $n$ to $1-k$. Equivalent since $1-k$ takes all integer values. Evaluate square and $n$-th roots if the answer is a rational number. Evaluate numerical quantities using approximate decimal values. Perform algebraic simplification. Example: $ln x = 2$ becomes $x = e^2$ Example: $log x = 2$ becomes $x = 100$ Example: $log(3,x) = 2$ becomes $x = 9$ Example: $10^(x+1) = 10^(2x)$ becomes $x+1 = 2x$ Example: $10^x = 3$ becomes $x = log 3$ Example: $e^x = 3$ becomes $x = ln 3$ Logarithms of negative numbers are not defined. Cramer's rule Evaluate a numerical determinant, or a symbolic one of dimension 2 or 3. Example: $x-1 = 2+y$ becomes $x - y = 1$ Example: $2x + 3 + x = 5$ becomes $3x + 3 = 5$ Align the terms in the same variable in the same column. You will be asked for the numbers of the two equations. You will be asked for the equation number and what to multiply by. You will be asked for the equation number and what to divide by. You will be asked for the equation numbers and multiplier. You will be asked for the two equation numbers Example: $y=1$, $x=2$ will be changed to $x=2, y=1$. Eliminate an equation that has simplified to an identity, such as 2=2. You will select a variable, which will subsequently be treated as constant. Example: if you have derived $x = 5$, $x = 2$, the equations cannot be satisfied. Push a nonnegative quantity inside absolute value. Push a nonnegative denominator inside absolute value. Push a nonnegative fraction inside absolute value. Solve a linear equation for the selected variable. You will be asked for the number of the equation that will change. You will be asked what to multiply the selected equation by. You will be asked what to divide the selected equation by. You will be asked for the multiplier and the target equation. You will be asked for the number of the other equation. You will be asked to select a variable. You will be asked for the number of the row that will change. You will be asked for the multiplier. You will be asked for the divisor. You will be asked for the multiplier and the other row number. You be asked for the number of the other row. Insert an identity matrix on the right (for calculating the matrix inverse). Example: $2x + 3y + x = 5$ becomes $3x + 3y = 5$. You will be asked to choose an equation number and then a variable. Perform algebraic simplifications. Example, $x + y = x + 2$ becomes $y = 2$ You will be asked to choose an equation and then to enter what to add. You will be asked to choose an equation and then to enter what to subtract. You will be asked to choose an equation and then to enter the divisor. When one equation is solved, you can use it to substitute in other equations. Example: if you have derived $x=2$ and $x=5$, the equations cannot be satisfied. write in matrix form You will be asked which two rows to swap. You will be asked for the numbers of the two rows. You will be asked for the number of the row and the multiplier. You will be asked for the number of the row and the divisor. You will be asked for two row numbers and the multiplier. Perform matrix multiplication. Use this if you have all zeroes in one column. Use this if you have all zeroes in one row. Use this if two rows are exactly the same. Use this if two rows are the same on the left, but not on the right. Convert an equation of one-column matrices to a system of equations. Perform matrix multiplication The matrix inverse will not be computed yet, just introduced symbolically. Compute the matrix inverse of a 2 by 2 matrix. Uses exact arithmetic and symbolic algebra. If it works the answer is exact. Works on a numerical matrix, using decimal arithmetic with limited accuracy. Drop absolute value signs around a nonnegative quantity. Example: $ |x-2| = x-2$, entering a new assumption $x\ge 2$. Example: |-2| = 2 Example: |2u| = 2|u| Example: |u/2| = |u|/2 Example: |x-1||x+1| = |(x-1)(x+1| Example: |(x-1)(x+1)| = |x-1||x+1| Example: |(x-1)/x| = |x-1| / |x| Example: |x^2-1| / |x-1| = |(x^2-1)/(x-1)| Example: |x|^4 =x^4 Example: |u^3|=|u|^3 If u is real, the absolute value on the right is unnecessary. Example: $|^3\sqrt u| = ^3\sqrt |u|$ Cancel, disregarding absolute value signs. Factor out the greatest common divisor of numerator and denominator. Example: |x|=2 becomes [x = 2, x = -2] Examples: |x|/x = x-2 becomes [x-2 = 1, x-2 = -1] Example: |x| < 2 becomes -2 < u < 2 Example: $|x| \le 2$ becomes $-2 \le u \le 2$ Example: 2 < |x| iff x < -2 or 2 < x Example: $2 \le |x|$ iff $x \le -2$ or $2 \le x$ Example: |x-1| = x-1 becomes $0 \le x-1$ Example: |x-1| = 1-x becomes $x-1 \le 0$ Example: $0 \le |x^2+1|$ is always true. Example: $-5 \le |x^2+1|$ is always true. Example: $-5 < |x^2+1|$ is always true. Example: |x^2+1| < 0 has no solution. Example: |x| < -5 has no solution. Example: $|x| \le -5$ has no solution. Example: $|x^3-x| \le -x^2$ becomes x^3-x = 0, and x=0 will be assumed. Example: |x^3-x| = -x^2 becomes x^3-x = 0, and x=0 will be assumed. Example: 2 > |x| becomes -2 < x < 2 Example: $2 \ge |x|$ becomes $-2 \le x \le 2$ Example: |x| > 2 iff -2 > x or x > 2 Example: $|x| \ge 2$ iff $-2 \ge x$ or $x \ge 2$ Example: $|x^2-1| \ge 0$ is true. Example: 0 > |x^2-1| has no solution. Example: -5 > |x| has no solution. Example: $-5 \ge |x|$ has no solution. Example: $-x^2 \ge |x^3-x|$ becomes x^3-x = 0, and x=0 will be assumed. Example: |x| > -5 is true Example: $|x| \ge -5$ is true Example: $-2 \le u \le 2$ becomes $|x| \le 2$ Example: x < -2 or 2 < x iff 2 < |x| Example: x^4 = |x|^4 Example: |u|^3 = |u^3| Example: 2 < x becomes x > 2 Example: x-2 < 5 becomes x<7. Select the 2. Example: x+2 < 5 becomes x=3. Select the 2. Example: -2 < -x becomes x < 2. Example: -x < - 2 becomes x > 2. Example: x/3 < 1 becomes x < 3. Select the 3. x/(x-1) < 2 becomes x(x-1) < 2(x-1)^2 when you select x-1. Example: 5x < 10 becomes x < 2. Select the 5. Produces 'No solution' or 'true', when the equality involves only numbers. Simplify an inequality of the form mentioned to 'true'. Simplify an inequality of the form mentioned to 'No solution'. u < v becomes u^2 < v^2 provided u is nonnegative. $0\le v$ will be derived or assumed. u < v becomes [u^2 < v^2, u<=0]. Use this if u can take negative values. Example: x<4 or x=4 becomes $x\le 4$. The "or" is implicit in bracket notation. Example: 1<x or 2<x becomes 1<x Use assumptions to reject or improve solutions to satisfy the original inequality. Example: 2 > x becomes x < 2 Example: -x > -2 becomes x < 2 Example: -2 > -x becomes x > 2 Example: x^2 > -1 is true Example: -1 > x^2 is false Example: 2 > x becomes [4 > x^2, x < 0] Example: [x > 2, x = 2] becomes $x \ge 2$ Example: $x \le 2$ becomes $2 \ge x$ Example: $x-2 \le 5$ becomes $x\le 7$. Select the 2. Example: $x+2 \le 5$ becomes x=3. Select the 2. Example: $-2 \le -x$ becomes $x \le 2$. Example: $x \le -2$ becomes $x \ge 2$. Example: $x/3 \le 1$ becomes $x \le 3$. Select the 3. Example: $x/(x-1) \le 2$ becomes $x(x-1) \le 2(x-1)^2$. Select x-1 Example: $x/5 \le 10$ becomes $x \le 2$. Select the 5. $u \le v$ becomes $u^2 \le v^2$ provided u is nonnegative. $0\le v$ will be derived or assumed. $u \le v$ becomes $u^2 \le v^2$ or $u\le 0$. Use this if u can take negative values. Example: $1\le x$ or $2\le x$ becomes $1\le x$ Example: $2 \ge x$ becomes $x \le 2$ Example: $-x \ge -2$ becomes $x \le 2$ Example: $-2 \ge -x$ becomes $x \ge 2$ Example: $x^2 \ge -1$ is true Example: $-1 \ge x^2$ is false Example: $2 \ge x$ becomes $[4 \ge x^2, x \le 0]$ Example: x^2 < 4 becomes |x| < 2 Example: x^2 < 4 becomes -2 < x < 2 Example: 4 < x^2 becomes 2 < |x| Example: 4 < x^2 becomes [x < -2, 2 < x] Example: 4 < x^2 < 9 becomes [-3 < x < -2, 2 < x < 3] Example: -2 < x^2 < 9 becomes x^2 < 9 Example: $-2 < x^2 \le 9$ becomes $x^2 \le 9$ Example: $\sqrt x < 2$ becomes $0 \le x < 4$ Example: $2\sqrt x < 2$ becomes $0 \le 4x < 4$ Example: $2 < \sqrt x$ becomes 4 < x Example: $x^2 < a => x < \sqrt a$ if $0\le x$ is already assumed. Example: $-1 < x^2$ is always true. Example: $x^2 < -1$ has no solution. Example: $-1 < \sqrt (x^2 - 1)$ becomes $0 \le x^2 -1$ Example: $x^2 \le 4$ becomes $|x| \le 2$ Example: $x^2 \le 4$ becomes $-2 \le x \le 2$ Example: $4 \le x^2$ becomes $2 \le |x|$ Example: $4 \le x^2$ becomes $[x \le -2, 2 \le x]$ Example: $4 \le x^2 \le 9$ becomes $[-3 \le x \le -2, 2 \le x \le 3]$ Example: $-2 \le x^2 \le 9$ becomes $x^2 \le 9$ Example: $-2 \le x^2 < 9$ becomes $x^2 < 9$ Example: $\sqrt x \le 2$ becomes $0 \le x \le 4$ Example: $2\sqrt x \le 2$ becomes $0 \le 4x \le 4$ Example: $2 \le \sqrt x$ becomes $4 \le x$ Example: $x^2 \le a => x \le \sqrt a$ if $0\le x$ is already assumed. Example: $-1 \le x^2$ is always true. Example: $x^2 \le -1$ has no solution. Example: $-1 \le sqrt(x^2 - 1)$ becomes $0 \le x^2 -1$ $1/x < a$ iff $x < 0$ or $1/a < x$, provided $a > 0$ $a < 1/x$ iff $0 < x < 1/a$ provided $a > 0$ $1/x < -a$ iff $-1/a < x < 0$ provided $a > 0$ $-a < 1/x$ iff $x < -1/a$ or $0 < x$ provided $a > 0$ Example: $1 < x < 2$ becomes $1/2 < x < 1$ Example: $1 < x \le 2$ becomes $1/2 \le x < 1$ Example: $-2 < 1/x < -1$ becomes $-1 < x < -1/2$ Example: $-2 < 1/x \le -1$ becomes $-1 \le x < -1/2$ Example: -2 < 1/x < 3 becomes [x < -1/2, 1/3 < x] Example: $-2 < 1/x \le 3$ becomes $[x < -1/2, 1/3 \le x]$ $1/x \le a$ iff x < 0 or $1/a \le x$, provided $a > 0$ $a \le 1/x$ iff $0 < x \le 1/a$ provided $a > 0$ $1/x \le -a$ iff $-1/a \le x < 0$ provided $a > 0$ $-a \le 1/x$ iff $x \le -1/a$ or 0 < x provided $a > 0$ Example: $1 \le 1/x < 2$ becomes $1/2 < x \le 1$ Example: $1 \le 1/x \le 2$ becomes $1/2 \le x \le 1$ Example: $-2 \le 1/x < -1$ becomes $-1 < x \le -1/2$ Example: $-2 \le 1/x \le -1$ becomes $-1 \le x \le -1/2$ Example: $-2 \le 1/x < 3$ becomes $[x \le -1/2, 1/3 < x]$ Example: $-2 \le 1/x \le 3$ becomes $[x \le -1/2, 1/3 \le x]$ Example: x^3 < 27 becomes x < 3 Example: x^4 < 16 becomes |x| < 2 Example: x^4 < 16 becomes -2 < x < 2 Example: 16 < x^4 becomes 2 < |x| Example: 16 < x^4 becomes [x < -2, 2 < x] Example: 16 < x^4 < 81 becomes [-3 < x < -2, 2 < x < 3] Example: $^4\sqrt x < 16$ becomes $0 \le x < 2$ Example: $^3\sqrt x < 2$ becomes x < 8 Example: $2 ^3\sqrt x < 1$ becomes 8x < 1 Example: $2 < ^3\sqrt x$ becomes 8 < x Example: x^4 < a becomes $x < ^4\sqrt a$ if $0\le x$ is already assumed. Example: $-1 < ^4\sqrt (x^2 - 1)$ becomes $0 \le x^2 -1$ Example: $x^3 \le 27$ becomes $x \le 3$ Example: $x^4 \le 16$ becomes $|x| \le 2$ Example: $x^4 \le 16$ becomes $-2 \le x \le 2$ Example: $16 \le x^4$ becomes $2 \le |x|$ Example: $16 \le x^4$ becomes $[x \le -2, 2 \le x]$ Example: $16 \le x^4 < 81$ becomes $[-3 \le x \le -2, 2 \le x \le 3]$ Example: $^4\sqrt x \le 16$ iff $0 \le x \le 2$ Example: $^3\sqrt x \le 2$ becomes $x \le 8$ Example: $2 ^3\sqrt x \le 1$ becomes $8x \le 1$ Example: $2 \le ^3\sqrt x$ becomes $8 \le x$ Example: $x^4 \le a$ becomes $x \le ^4\sqrt a$ if $0\le x$ is already assumed. Example: $-1 \le ^4\sqrt (x^2 - 1)$ becomes $0 \le x^2 -1$ Example: 0 < x(x^2+1) becomes 0 < x Example: $0 < 1/\sqrt x$ becomes $0 < \sqrt x$ Example: $0 < x/\sqrt (x-1)$ becomes 0 < x(x-1) Example: 0 < (x-1)/(x-2) becomes 0 < (x-1)(x-2) Example: $1/\sqrt x < 0$ becomes $\sqrt x < 0$ Example: $x/\sqrt (x-1) < 0$ becomes $x(x-1) < 0$ $ax \pm b < 0$ iff $a(x\pm b/a) < 0$ u < v => v > u Example: (x-1)(x+1) < 0 iff -1 < x < 1. Also handles more factors. Example: 0 < (x-1)(x+1) iff x < -1 or 1 < x. Also handles more factors. Example: $0 \le x(x^2+1)$ becomes $0 \le x$ Example: $0 \le 1/\sqrt x$ becomes $0 \le \sqrt x$ Example: $0 \le x/\sqrt (x-1)$ becomes $0 \le x(x-1)$ Example: $0 \le (x-1)/(x-2)$ becomes $0 \le (x-1)(x-2)$ Example: $1/\sqrt x \le 0$ becomes $\sqrt x \le 0$ Example: $x/\sqrt (x-1) \le 0 $becomes $x(x-1) \le 0$ $ax \pm b \le 0$ iff $a(x\pm b/a) \le 0$ $u \le v => v \le u$ Example: $(x-1)(x+1) \le 0$ iff $-1 \le x \le 1$. Also handles more factors. Example: $0 \le (x-1)(x+1)$ iff $x \le -1 or 1 \le x$. Also handles more factors. Example: 4 > x^2 becomes 2 > |x| Example: 4 > x^2 becomes -2 < x < 2 Example: x^2 > 4 becomes |x| > 2 Example: x^2 > 4 becomes [x < -2, x > 2] Example: $2 > \sqrt x$ becomes $0 \le x < 4$ Example: $2 > 2\sqrt x < 2$ becomes $0 \le 4x < 4$ Example: $\sqrt x > 2$ becomes x > 4 Example: 4 > x^2 becomes 2 > x if $0\le x$ is already assumed. Example: $x^2 > -1$ is always true. Example: $-1 > x^2$ has no solution. Example: $\sqrt (x^2-1) > -1$ becomes $x^2-1 \ge 0$ Example: $4 \ge x^2$ becomes $2 \ge |x|$ Example: $4 \ge x^2$ becomes $-2 \le x \le 2$ Example: $x^2 \ge 4$ becomes $|x| \ge 2$ Example: $x^2 \ge 4$ becomes $[x \le -2, 2 \le x]$ Example: $2 \ge \sqrt x$ becomes $0 \le x \le 4$ Example: $2 \ge 2\sqrt x$ becomes $0 \le 4x \le 4$ Example: $\sqrt x \ge 2$ becomes $x \ge 4$ Example: $4 \ge x^2$ => $2 \ge x$ if $0\le x$ is already assumed. Example: $x^2 \ge -1$ is always true. Example: $-1 \ge x^2$ has no solution. Example: $\sqrt (x^2-1) \ge -1$ becomes $x^2-1 \ge 0$ a > 1/x iff x<0 or x > 1/a, provided $a > 0$ $1/x > a$ iff $0 < x < 1/a$, provided $a > 0$ $-a > 1/x$ iff $-1/a < x < 0$, provided $a > 0$ $1/x > -a$ iff $x < -1/a$ or $x > 0$, provided $a > 0$ $a \ge 1/x$ iff x<0 or $x \ge 1/a$, provided a > 0 $1/x \ge a$ iff $0 < x \le 1/a$, provided a > 0 $-a \ge 1/x$ iff $-1/a \le x < 0$, provided a > 0 $1/x \ge -a$ iff $x \le -1/a$ or x > 0, provided a > 0 Example: 27 > x^3 becomes $3 > x$ Example: 16 > x^4 becomes $2 > |x|$ Example: 16 > x^4 becomes $-2 < x < 2$ Example: x^4 > 16 becomes |x| > 2 Example: x^4 > 16 becomes [-2 > x, x > 2] Example: $2 > ^3\sqrt x$ becomes 8 > x Example: $1 > 2 ^3\sqrt x$ becomes 1 > 8x Example: $^3\sqrt x > 2$ becomes x > 8 Example: $2 > ^3\sqrt x$ becomes 8 > x Example: $a > x^4$ becomes $^4\sqrt a > x$ if $0\le x$ is already assumed. Example: $^4\sqrt (x^2 - 1) > -1$ becomes $x^2 -1 \ge 0$ Example: $27 \ge x^3$ becomes $3 \ge x$ Example: $16 \ge x^4$ becomes $2 \ge |x|$ Example: $16 \ge x^4$ becomes $-2 \le x \le 2$ Example: $x^4 \ge 16$ becomes $|x| \ge 2$ Example: $x^4 \ge 16$ becomes $[-2 \ge x, x \ge 2]$ Example: $2 \ge ^3\sqrt x$ becomes $8 \ge x$ Example: $1 \ge 2 ^3\sqrt x$ becomes $1 \ge 8x$ Example: $^3\sqrt x \ge 2$ becomes $x \ge 8$ Example: $^4\sqrt (x^2 - 1) \ge -1$ becomes $x^2 -1 \ge 0$ Example: $1/\sqrt x > 0$ becomes $\sqrt x > 0$ Example: $x/\sqrt (x-1) > 0$ becomes x(x-1) > 0 Example: (x-1)/(x-2) > 0 becomes (x-1)(x-2) > 0 Example: $0 > 1/\sqrt x$ becomes $0 > \sqrt x$ Example: $0 > x/\sqrt (x-1)$ becomes 0 > x(x-1) $0 > ax \pm b$ iff $0 > a(x\pm b/a)$ Example: 0 > (x-1)(x+1) iff -1 < x < 1. Also handles more factors. Example: (x-1)(x+1) > 0 iff x < -1 or 1 < x. Also handles more factors. Example: $1/\sqrt x \ge 0$ becomes $\sqrt x \ge 0$ Example: $x/\sqrt (x-1) \ge 0$ becomes $x(x-1) \ge 0$ Example: $(x-1)/(x-2) \ge 0$ becomes $(x-1)(x-2) \ge 0$ Example: $0 \ge 1/\sqrt x$ becomes $0 \ge \sqrt x$ Example: $0 \ge x/\sqrt (x-1)$ becomes $0 \ge x(x-1)$ $0 \ge ax \pm b$ iff $0 \ge a(x\pm b/a)$ Example: $0 \ge (x-1)(x+1)$ iff $-1 \le x \le 1$. Also handles more factors. Example: $(x-1)(x+1) \ge 0$ iff $x \le -1$ or $1 \le x$. Also handles more factors. Expands all the way, does not use sigma notation. Can create terms. Expands using sigma notation and binomial coefficients. Express binomial coefficients using factorials. Use the definition of factorial as a product. Does not multiply it out. Compute the value of a factorial. Example: 6! = 720. Evaluate a specific binomial coefficient. Example: (4 2) = 6 Express $\sum $ using +. The sum must have a constant number of terms. If each term is a number, evaluate using exact rational arithmetic. Example: $7! = 7\times 6!$ Example: $7!/7 = 6!$ Example: $7!/6! = 7$ Example: $n!/(n-2)! = n(n-1)$ Example: $7/7! = 1/6!$ Example: $6!/7! = 1/7$ Example: $(n-2)!/n! = 1/(n(n-1))$ Factor the cube of a sum. Factor the cube of a difference. Factor the fourth power of a sum. Factor the fourth power of a difference. Factor a power of a sum. Factor a power of a difference. Example: the sum of 1 from 1 to 10 is 10. Pull a minus sign out from an indexed sum. Pull a constant out of an indexed sum. Break an indexed sum into two (or more) sums. Example: the sum of $i$ for $i = 1$ to 100 is 100(101)/2 = 5050. Formula for the sum of the first n perfect squares. The sum of x^i for $i=0$ to n has this elegant closed form. You will be asked how many terms to write out explicitly. Specify a parameter value and evaluate using exact rational arithmetic. Specify a parameter value and evaluate using (inexact) decimal arithmetic. Evaluate a numerical sum using exact arithmetic. No parameters allowed. Evaluate a numerical sum using decimal arithmetic. No parameters allowed. Express the summand as a polynomial in the index variable, if possible. Example: the sum of 1/(k+1) - 1/k from 1 to n becomes 1/(n+1) - 1 Example: change a sum from k=0 to n to a sum from k = 1 to n+1 Before multiplying out a product of sums you may need to rename a variable. Convert a product of sums to a double sum using the distributive law. Example: Change a sum from 1 to n+1 to a sum from 1 to n, plus the last term. The formula for the sum of the first n cubes The formula for the sum of the first n fourth powers Push a derivative into an indexed sum Pull a derivative out of an indexed sum Push an integral into an indexed sum Pull an integral out of an index sum Push a constant into an indexed sum or series. Write an indexed sum as a difference of two sums with zero as the starting index of summation. Write an indexed sum as a difference of two sums with a new, specified starting index. You will be asked to choose the induction variable. You will be asked for the starting value of the induction variable. Assume the induction hypothesis and state what is to be proved. Use the induction hypothesis to simplify the current line. Use this when the induction step is completed, to draw the final conclusion. Simplify an inequality of the stated form to true. Simplify an inequality of the stated form to true. Example: $sin x^2 \le x^2$. u < v iff ln u < ln v, provided u > 0. u < v iff log u < log v, provided u > 0. Example: 2 < ln x becomes e^2 < x Example: ln x < 2 becomes x < e^2 Example: 2 < log x becomes 10^2 < x Example: log x < 2 becomes x < 10^2 You will specify the number ? to use as the base of exponents. $u \le v$ iff $ln u \le ln v$, provided u > 0. $u \le v$ iff $log u \le log v$, provided u >0. Example: $2 \le ln x$ becomes $e^2 \le x$ Example: $ln x \le 2$ becomes $x \le e^2$. Example: $2 \le log x$ becomes $10^2 \le x$. Example: $log x \le 2$ becomes $x \le 10^2$. u > v iff ln u > ln v, provided u > 0. u > v iff log u > log v, provided u > 0. Example: ln x > 2 becomes x > e^2. Example: 2 > ln x becomes e^2 > x. Example: log x > 2 becomes x > 10^2. Example: 2 > log x becomes 10^2 > x. $u \ge v$ iff $ln u \ge ln v$, provided u > 0 $u \ge v$ iff $log u \ge log v$, provided u >0 Example: $ln x \ge 2$ becomes $x \ge e^2$. Example: $2 \ge ln x$ becomes $e^2 \ge x$. Example: $log x \ge 2$ becomes $x \ge 10^2$. Example: $2 \ge log x$ becomes $10^2 \ge x$. Example: $n < 2^n$ for $n > M$, for a specific but unspecified number $M$ Example: $ln n < \sqrt n$ for $n > M$, for a specific but unspecified number $M$ Example: 10^(log 3x) becomes 3x. Example: log 100 becomes 2 The log of 1 is zero since 10^0 = 1. The log of 10 is 1, since 10^1 = 1. Convert logarithms base 10 to natural logarithms. Express a power using base 10 and a log in the exponent. Example: $400 = 10^2\times 4$. Does not factor completely, only takes out tens. Example: 10^(2 log x) becomes x^2. Example: $log (4/5) = - log (5/4)$ Example: $log(3,4/5) = - log(3, 5/4)$ Example: log x^3 = 3 log x Example: log 3x = log 3 + log x Example: log 1/2 = -log 2 Example: log x/2 = log x - log 2 Example: log 2 + log x = log 2x Example: log x - log 2 = log a/2 Example: log x + log 2 - log 3 =log 2x/3 Example: 2 log x = log x^2 Example: $log \sqrt 3 = \onehalf log 3$ Example: $log ^3\sqrt x = (1/3) log x$ The log of 1 is 0 since 10^0 = 1. Example: $400 = 10^2\times 4$. Does not factor completely, only takes out tens. You will be asked to enter a. Example: log x = $\onehalf log u^2$ Evaluate logs using decimal approximations. Convert base 10 logarithms to natural logarithms. This fundamental law connects natural logs and the exponential function. In words: e is the base of natural logarithms. The natural log of 1 is 0, since e^0 = 1. Example: ln e^2 = 2 Express an arbitrary power using a power of $e$ and a natural logarithm. Eliminate a natural log in an exponent of $e$. Example: ln x^2 = 2 ln x Example: ln 2x = ln 2 + ln x Example: ln 1/2 = -ln 2 Example: ln x/2 = ln x - ln 2 Example: ln (x-1) + ln (x+1) = ln (x-1)(x+1) Example: ln x - ln 2 = ln x/2 Example: ln x + ln 2 - ln 3 = ln (2x/3) Example: 2 ln x = ln x^2 Example: $ln \sqrt 3 = \onehalf ln 3$ Example: $ln ^3\sqrt x = (1/3) ln x$ You will be asked to enter a. Example: ln (1 + 1/n) = 1/n ln(1+1/n)^n Evaluate natural logarithms using decimal approximations. Example: $ln (4/5) = - ln (5/4)$ Example: $sin x cos(\pi /2) + cos x sin(\pi /2) = sin(x+\pi /2)$ Example: $sin x cos(\pi /2) - cos x sin(\pi /2) = sin(x-\pi /2)$ Example: $cos x cos(\pi /2) - sin x sin(\pi /2) = cos(x+\pi /2)$ Example: $cos x cos(\pi /2) + sin x sin(\pi /2) = cos(x-\pi /2)$ Example: (sin 4u)/(1+cos 4u) = tan 2u Example: (1-cos 4u)/sin 4u = tan 2u Example: (1+cos 4u)/sin 4u = cot 2u Example: (sin 4u)/(1-cos 4u) = cot 2u Example: $(tan x + tan \pi /2)/(1-tan x tan \pi /2) = tan(x+\pi /2)$ Example: $(tan x - tan \pi /2)/(1+tan x tan \pi /2) = tan(x-\pi /2)$ Example: $(cot x cot(\pi /4) - 1)/(cot x + cot \pi /4) = cot(x+\pi /4)$ Example: $(1 + cot x cot \pi /4)/(cot \pi /4 - cot x) = cot(x-\pi /4)$ Example: $1-cos(\pi /3)$ becomes $2sin^2 \pi /6$ Convert x + iy to polar form $r e^(i\theta )$. Express a complex exponential in terms of cosine and sine. Since $e^(i\theta )$ lies on the unit circle, its absolute value is 1. Since $Re^(i\theta )$ lies on the circle of radius R, its absolute value is R. If the sign of R is unknown, you need the absolute value on the right. Example: $-2 = 2e^(i\pi )$ Example: $^3\sqrt (-2) = e^(\pi i/3) ^3\sqrt 2$ Example: 2/(3e^t) = 2e^(-t)/3 Example: x^3 = 1 becomes $x = e^(2k\pi i/3)$ Example: $x = e^(2k\pi i/3)$ becomes $[x=1, x=e^(2\pi i/3), x=e^(4\pi i/3)]$ Example: 2^(log(2,3)) = 3 The log to the base b of b is 1. The log to the base b of b^n is n. Example: log 2x = log 2 + log x Example: $log (\onehalf ) = -log 2$ The log to any base of 1 is zero, since b^0 = 1. After using this, you will be able to change the base. Example: $log(3^2,x) = \onehalf log (3,x)$ Example: log x^2 = 2 log x Example: $log(2, 84) = log(2,2^2\times 21)$ Example: log x - log 2 = log x/2 Example: 5^(2 log(5,x)) becomes x^2 Convert logarithms base b to natural logarithms Convert logarithms base b to logarithms base 10 Convert logarithms base b to logarithms base a Example: log(3^2,x) = (1/2) log (3,x) Definition of log In words: e is the base of natural logarithms. Convert natural logarithms to logarithms base 10. Example: x^5 becomes 3^5 log(3,x) sin 0 = 0 cos 0 = 1 tan 0 = 0 Sine is zero at multiples of $\pi $. Cosine is 1 at even multiples of $2\pi $. Tangent is zero at multiples of $\pi $. Example: $sin 370\deg = sin 10\deg $ Example: $sin 9\pi /4 = sin \pi /4$ Examples: $sin 3\pi /2 = -1; cos 180\deg = -1; cot 90\deg = 0$. Examples: $sin 30\deg = 1/2; cos \pi /3 = 1/2; tan 2\pi /3 = -\sqrt 3$. Examples: $sin 45\deg = 1/\sqrt 2; tan 3\pi /4 = -1$. $\pi $ radians = 180 degrees = half a circle of arc 180 degrees = $\pi $ radians = half a circle of arc Example: $15\deg = 45\deg - 30\deg $. Use this to evaluate $sin 15\deg $ exactly. Evaluate trig functions using decimal approximations. Express tan in terms of sin and cos Express cot in terms of tan Express cot in terms of sin and cos Definition of sec Definition of csc Definition of tan Definition of cot The reciprocal of sine is the cosecant. The reciprocal of cosine is the secant The reciprocal of the tangent is the cotangent The reciprocal of the tangent can be expressed in terms of sin and cos. The reciprocal of the cotangent is the tangent The reciprocal of the cotangent can be expressed in terms of sin and cos. The reciprocal of the secant is the cosine The reciprocal of the cosecant is the sine. The reciprocal of the sine is the cosecant Express tan in terms of cot This fundamental identity is the Pythagorean theorem in disguise. Use this form of $sin^2 u + cos^2 u = 1$ to simplify $1 - sin^2 u$. Use this form of $sin^2 u + cos^2 u = 1$ to simplify $1 - cos^2 u$. Express $sin^2$ in terms of $cos^2$. Express $cos^2$ in terms of $sin^2$. To remember this identity, divide $sin^2 + cos^2 = 1$ by $cos^2$. Use this to simplify $tan^2 u + 1$. Use this to simplify $sec^2 u - 1$. Express $sec^2$ in terms of $tan^2$. Express $tan^2$ in terms of $sec^2$. Example: $sin^5 t = sin t (1-cos^2 t)^2$ Example: $cos^5 t = cos t (1-sin^2 t)^2$ Example: $tan^5 t = tan (sec^2 t-1)^2$ Example: $sec^5 t = sec t (tan^2 t+1)^2$ Example: (1-cos t)^2(1+cos t)^2 = sin^4 t Example: (1-sin t)^2(1+sin t)^2 = cos^4 t To remember this identity, divide $sin^2 + cos^2 = 1 by sin^2$. Use this to simplify $cot^2 u + 1$. Use this to simplify $csc^2 u - 1$. Express $csc^2$ in terms of $cot^2$. Express $cot^2$ in terms of $csc^2$. Example: $csc \pi /6 = sec \pi /3$ Example: $cot \pi /6 = tan \pi /3$ Example: $cot^5 t = cot (csc^2 t-1)^2$ Example: $csc^5 t = csc t (cot^2 t+1)^2$ Example: $sin(x+\pi /4)= sin x cos \pi /4 + cos x sin \pi /4$ Example: $sin(x-\pi /4)= sin x cos \pi /4 - cos x sin \pi /4$ Example: $cos(x+\pi /4)= cos x cos \pi /4 - sin x sin \pi /4$ Example: $cos(x-\pi /4)= cos x cos \pi /4 + sin x sin \pi /4$ Example: $tan(x+\pi /4)=(tan x+tan \pi /4)/(1-tan x tan \pi /4)$ Example: $tan(x-\pi /4)=(tan x-tan \pi /4)/(1+tan x tan \pi /4)$ Example: $cot(x+\pi /4)=(cot x cot \pi /4-1)/(cot x+cot \pi /4)$ Example: $cot(x-\pi /4)=(1+cot x cot \pi /4)/(cot \pi /4-cot x)$ Examples: sin 4x = 2 sin 2x cos 2x; $sin 40\deg = 2 sin 20\deg sin 20\deg $ Examples: cos 4x = cos^2 x - sin^2 x; $cos 40\deg = cos^2 20\deg - sin^2 20\deg $ Express $cos 2\theta $ in terms of $sin^2 \theta $. Express $cos 2\theta $ in terms of $cos^2 \theta $. Express $tan 2\theta $ in terms of $tan \theta $. Express $cot 2\theta $ in terms of $cot \theta $. Express $sin \theta cos \theta $ in terms of $sin 2\theta $ Express $2 sin \theta cos \theta $ in terms of $sin 2\theta $ Express $cos^2 \theta - sin^2 \theta $ as a single trig function, $cos(2\theta )$ Use this to get rid of $sin^2$ in favor of a single trig function. Use this to get rid of $cos^2$ in favor of a single trig function. Example: $3\theta = 2\theta + \theta $ Example: $7\theta = 3\theta + 4\theta $; you enter the 3 when you are asked for it. This triple-angle formula can save you several steps. Example: $sin 7\theta = -sin^7 \theta + 21 cos^2 \theta sin^5 \theta + ...$ Example: $cos 7\theta = cos^7 \theta - 21 cos^5 \theta sin^2 \theta + ...$ Example: x/3 = 3/4 becomes 4x = 9 Example: 3 = x becomes x = 3 The specified term will be moved from the left to the right side. The specified term will be moved from the right to the left side. Add a specified term to both sides Subtract a specified term from both sides Multiply both sides by a specified term. Example: $1 - sin^2 x + tan x = tan x + cos^2 x$ becomes $1-sin^2 x = cos^2 x$. Example: $\sqrt (1-sin^2 x) = cos x$ becomes $1-sin^2 x = cos^2 x$. Example: tan^2 x = sin^2 x / cos^2 x becomes tan x = sin x / cos x Example: tan^3 x = sin^3 x / cos^3 x becomes tan x = sin x / cos x You will be asked what function to apply. Use this to disprove a false identity or to test one you can't verify. These angles are $30\deg $ above the plus and minus x-axes. These angles are $30\deg $ below the plus and minus x-axes. These angles are the multiples of $60\deg $ above the x-axis. These angles are the multiples of $60\deg $ below the x-axis. That is, plus or minus $30\deg $. That is, plus or minus $30\deg $ from the negative x-axis. That is, plus or minus $60\deg $. That is, plus or minus $120\deg $. That is, $30\deg $ plus multiples of $\pi $ (not $2\pi $, note $210\deg $ is included). That is, $-30\deg $ plus multiples of $\pi $ (not $2\pi $, note $150\deg $ is included). That is, $60\deg $ plus multiples of $\pi $ (not $2\pi $, note $240\deg $ is included). That is, $-60\deg $ plus multiples of $\pi $ (not $2\pi $, note $120\deg $ is included). These angles are $45\deg $ up from the plus and minus x-axes. These angles are $45\deg $ down from the plus and minus x-axes. These angles are $45\deg $ right from the plus and minus y-axes. These angles are $45\deg $ left from the plus and minus y-axes. That is, $45\deg $ plus multiples of $\pi $ (not $2\pi $, note $225\deg $ is included). That is, $-45\deg $ plus multiples of $\pi $ (not $2\pi $, note $135\deg $ is included). sin u is zero at multiples of $\pi $. sin u is 1 when u is $\pi /2$ plus a multiple of $2\pi $. sin u is -1 when u is $3\pi /2$ plus a multiple of $2\pi $. cos u is 0 when u is an odd multiple of $\pi /2$. cos u = 1 when u is a multiple of $2\pi $. cos u = -1 when u is an odd multiple of $\pi $. Example: $tan x^2 = 0$ becomes $sin x^2 = 0$. Example: $cot x^2 = 0$ becomes $cos x^2 = 0$. Example: sin x = 3/4 becomes $x = (-1)^n arcsin 3/4 + n\pi $ Exmaple: sin x = 3/4 becomes $[x = arcsin 3/4 + 2n\pi , x = -arcsin 3/4 + (2n+1)\pi ]$ Example: cos x = 3/4 becomes $[x = arccos 3/4+2n\pi , x = -arccos 3/4 + 2n\pi ]$ Example: tan x = 3 becomes $x = arctan 3 + n\pi $ Example: $arcsin(\onehalf ) = \pi /6$. Only a few values will evaluate exactly. Example: $arccos(\onehalf ) = \pi /3$. Only a few values will evaluate exactly. Example: $arctan 1 = \pi /4$. Only a few values will evaluate exactly. If cot z = x then tan z = 1/x. If sec z = x then cos z = 1/x. If csc z = x then sin z = 1/x. arcsin is an odd function arccos is not quite odd but obeys this similar identity. arctan is an odd function. Put the solutions in the form $c + 2n\pi $, if $2\pi $ is the period. Example: sin u = 2 has no solution. Example: cos u = 2 has no solution. If $sin \theta = x$ then $tan \theta = x/\sqrt (1-x^2)$. If $cos \theta = x$ then $tan \theta = \sqrt (1-x^2)/x$. The defining property of arctan. The defining property of arcsin. If $cos \theta = x$ then $sin \theta = \sqrt (1-x^2)$. If $tan \theta = x$ then $sin \theta = x/\sqrt (x^2+1)$. If $sin \theta = x$ then $cos \theta = \sqrt (1-x^2)$ The defining property of arccos If $tan \theta = x$ then $cos \theta = 1/\sqrt (x^2+1)$ If $sin \theta = x$ then $sec \theta = 1/\sqrt (1-x^2)$ If $cos \theta = x$ then $sec \theta = 1/x$ If $tan \theta = x$ then $sec \theta = \sqrt (x^2+1)$ Example: $arctan (tan \pi /3) = \pi /3$ Example: $arcsin(sin \pi /3) = \pi /3$ Example: $arccos(cos \pi /5) = \pi /5$ c1 is constant on intervals where tan x is defined, a constant of integration. The angle whose sin is x and the angle whose cosine is x are complementary. That is, the sum is $\pm \pi /2$, depending on the sign of x. Cosine is the sine of the complement. Sine is the cosine of the complement. Cotangent is the tangent of the complement. Tangent is the cotangent of the complement. Cosecant is the secant of the complement. Secand is the cosecant of the complement. Example: $sin (\pi /3) = cos (\pi /6)$ Example: $cos (\pi /3) = sin (\pi /6)$ Example: $tan (\pi /3) = sin (\pi /6)$ Example: $cot (\pi /3) = tan (\pi /6)$ Example: $sec (\pi /3) = csc (\pi /6)$ Example: $csc (\pi /3) = sec (\pi /6)$ Example: $sin (30\deg ) = cos (60\deg )$ Example: $cos (30\deg ) = sin (60\deg )$ Example: $tan (30\deg ) = sin (60\deg )$ Example: $cot (30\deg ) = tan (60\deg )$ Example: $sec (30\deg ) = csc (60\deg )$ Example: $csc (30\deg ) = sec (60\deg )$ Example: $15\deg +10\deg = (15+10)\deg = 25\deg $. Only numbers can be directly added. Example: $2\times 30\deg = (2\times 30)\deg = 60\deg $ Example: $60\deg /2 = (30)\deg $ sin is an odd function. cos is an even function. tan is an odd function. cot is an odd function. sec is an even function. csc is an odd function. sin^2 is an even function. cos^2 is an even function. tan^2 is an even function. cot^2 is an even function. sec^2 is an even function. csc^2 is an even function. sin is periodic with period $2\pi $. Example: $sin (9\pi /4) = sin (\pi /4)$ cos is periodic with period $2\pi $. Example: $cos (9\pi /4) = cos (\pi /4)$ tan is periodic with period $\pi $. Example: $tan (3\pi /4) = tan (\pi /4)$ sec is periodic with period $2\pi $. Example: $sec (9\pi /4) = sec (\pi /4)$ csc is periodic with period $2\pi $. Example: $csc (9\pi /4) = csc (\pi /4)$ cot is periodic with period $\pi $. Example: $cot (3\pi /4) = cot (\pi /4)$ sin^2 is periodic with period $\pi $. Example: $sin^2 (3\pi /4) = sin^2 (\pi /4)$ cos^2 is periodic with period $\pi $. Example: $cos^2 (3\pi /4) = cos^2 (\pi /4)$ sec^2 is periodic with period $\pi $. Example: $sec^2 (3\pi /4) = sec^2 (\pi /4)$ csc^2 is periodic with period $\pi $. Example: $csc^2 (3\pi /4) = csc^2 (\pi /4)$ Example: $sin 200\deg = -sin 20\deg $ Example: $sin 160\deg = sin 20\deg $ Example: $cos 200\deg = -cos 20\deg $ Example: $cos 160\deg = -cos 20\deg $ Express $sin^2$ in terms of a single trig function instead of a power. Express $cos^2$ in terms of a single trig function instead of a power. Change a product of trig functions into a single trig function. There are two formulas for $tan (\theta /2)$. Choose the best one by context. There are two formulas for $cot (\theta /2)$. Choose the best one by context. Express $sin(\theta /2)$ in terms of $cos \theta $ Express $cos(\theta /2)$ in terms of $cos \theta $ Example: $60\deg = 2\times 30\deg $. The reverse of the double angle formula. Example: $sin (x+\pi /4) cos (x-\pi /4) = \onehalf [sin(2x)+sin(\pi /2)]$ Example: $cos (x+\pi /4) sin (x-\pi /4) = \onehalf [sin(2x)-sin(\pi /2)]$ Example: $sin (x+\pi /4) sin (x-\pi /4) = \onehalf [cos(\pi /2)-cos(2x)]$ Example: $cos (x+\pi /4) cos (x-\pi /4) = \onehalf [cos(2x)+cos(\pi /2)]$ Write a sum of sines can be as a product of a sine and a cosine. Write a difference of sines as a product of a sine and a cosine. Write a sum of cosines as a product of a sine and a cosine. 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Sindbad File Manager Version 1.0, Coded By Sindbad EG ~ The Terrorists