Precalc +: Inverses

One-to-one function

Function in which each output has a unique input (passes the horizontal line test)

What must you do when finding the inverse of a periodic function?

Restrict the domain to a set of values that covers the entire range of the original function

Inverse periodics with domain D: (-π/2, π/2)

sin^-1, tan^-1, csc^-1

Inverse periodics with domain D: (0, π)

cos^-1, sec^-1, cot^-1

Quadrants the ouput of inverse periodics with domain D: (-π/2, π/2) can be in

Q1 (positive) + -Q4 (negative)

Quadrants the ouput of inverse periodics with domain D: (0, π) can be in

Q1 (positive) + Q2 (negative)

Strategy for evaluating inverse trig expressions

* Determine the family whose ouput matches the function and the input
* Determine the quadrant the answer will be in (depending on the sign of the input)
* Write the angle in the correct quadrant

Strategy for evaluating compositions of trig functions

* Work from inside out
* Show your answer for BOTH problems

Strategy for evaluating compositions of trig functions with non-special angles (trig(inv trig(ratio))

* Determine the quadrant the inverse trig function ouput will be in
* Draw a triangle in that quadrant
* Label two sides of the triangle with the numbers in the ratio (ex. cos^-1 (-3/5), label the adjacent -3, label the hypotenuse 5)
* Solve for the last side with pythagorean equation
* Use the triangle to solve the outside function

Strategy for evaluating compositions of trig functions with non-special angles (inv trig(trig(angle))

* Determine the quadrant the given angle is in and the sign the ratio will have
* Using the sine of the ouput of the periodic function, determine the quadrant the final answer will be in
* Readjust the angle to the correct quadrant
* For cofunctions, determine the reference angle of the periodic function, subtract that angle from π/2, and then put that angle in the correct quadrant

Solving with a calculator (3 basic functions, inverse and normal)

Use the button lol

Reciprocal functions

* For periodics: Find the reciprocal of the function (ex. sec(x) = 1/cos(x))
* For inverses: Flip the number in the parentheses (ex. sec^-1(x) = cos^-1 (1/x))

Cotangent and tangent with a calculator

Ensure your final answer matches the quadrant that should have been given by the original problem (ex. cot^-1 (-1/2) should be in Q2, but your calculator will give you Q4)