Inverse Trig Functions Lesson 3.5

Start with Sine Function x y = sin(x) -3.1416 0.0000 -2.6180 -0.5000 -2.0944 -0.8660 -1.5708 -1.0000 -1.0472 -0.5236 0.5236 0.5000 1.0472 0.8660 1.5708 1.0000 2.0944 2.6180 3.1416 3.6652 4.1888 4.7124 5.2360 5.7596 6.2832 Given y = sin (x) Table of values Graph What if we reversed the ordered pairs … y for x ?

Reversed Ordered Pairs x y 0.0000 -3.1416 -0.5000 -2.6180 -0.8660 -2.0944 -1.0000 -1.5708 -1.0472 -0.5236 0.5000 0.5236 0.8660 1.0472 1.0000 1.5708 2.0944 2.6180 3.1416 3.6652 4.1888 4.7124 5.2360 5.7596 6.2832 Problem This is not a function Fails the vertical line test There are multiple (x,y)'s where x = .5 Solution Limit the range

The Inverse Trig Function We say Similarly for inverse cosine The range of cos-1x is limited differently Note pg 258 for domain, range of other functions

Evaluating Inverse Functions Consider cos-1(-0.5) We are asking what angle has a cosine value of -0.5 Cosine negative in quadrants 2 and 3 But for cos-1(x) we look only in 1 & 2 Calculator also capable of evaluating inverse trig functions 2 -1

Note: newer calculators will have these functions – find in Catalog Try It Out Consider these Note: newer calculators will have these functions – find in Catalog

Composition of Trig Functions and Inverses Recall that in general f-1(f(x)) = f(f-1(x)) = x For trig functions this is the same sin(arcsin(x)) = arcsin(sin(x)) The restriction on the domain and range of the inverse functions must apply Thus sin-1(sin(3)) could not be 3 Note calculator results

Composition of Trig Functions and Inverses Try these … with and without calculator

Solving Inverse Trig Equations Given Strategy Isolate the sin-1x Take the sine of both sides of the equation

Try it Out Try this one

Assignment Lesson 3.5 Page 265 Exercises 1 – 65 EOO