Inverse Trig Functions Lesson 3 5 Start with

Start with Sine Function Given y = sin (x) ¡ Table of values ¡

Reversed Ordered Pairs ¡ Problem l l l ¡ This is not a function

The Inverse Trig Function ¡ We say ¡ Similarly for inverse cosine ¡ The

Evaluating Inverse Functions ¡ Consider cos-1(-0. 5) l ¡ We are asking what angle

Try It Out ¡ Consider these Note: newer calculators will have these functions –

Composition of Trig Functions and Inverses ¡ Recall that in general l f-1(f(x)) ¡

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

Solving Inverse Trig Equations ¡ Given ¡ Strategy l Isolate the sin-1 x l

Assignment Lesson 3. 5 ¡ Page 265 ¡ Exercises 1 – 65 EOO ¡

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Inverse Trig Functions Lesson 3. 5

Start with Sine Function Given y = sin (x) ¡ Table of values ¡ Graph What if we reversed ¡ the ordered pairs … y for x ? x y = sin(x) -3. 1416 0. 0000 -2. 6180 -0. 5000 -2. 0944 -0. 8660 -1. 5708 -1. 0000 -1. 0472 -0. 8660 -0. 5236 -0. 5000 0. 0000 0. 5236 0. 5000 1. 0472 0. 8660 1. 5708 1. 0000 2. 0944 0. 8660 2. 6180 0. 5000 3. 1416 0. 0000 3. 6652 -0. 5000 4. 1888 -0. 8660 4. 7124 -1. 0000 5. 2360 -0. 8660 5. 7596 -0. 5000 6. 2832 0. 0000 2

Reversed Ordered Pairs ¡ Problem l l l ¡ This is not a function Fails the vertical line test There are multiple (x, y)'s where x =. 5 Solution l l Limit the range x y 0. 0000 -3. 1416 -0. 5000 -2. 6180 -0. 8660 -2. 0944 -1. 0000 -1. 5708 -0. 8660 -1. 0472 -0. 5000 -0. 5236 0. 0000 0. 5000 0. 5236 0. 8660 1. 0472 1. 0000 1. 5708 0. 8660 2. 0944 0. 5000 2. 6180 0. 0000 3. 1416 -0. 5000 3. 6652 -0. 8660 4. 1888 -1. 0000 4. 7124 -0. 8660 5. 2360 -0. 5000 5. 7596 0. 0000 6. 2832 3

The Inverse Trig Function ¡ We say ¡ Similarly for inverse cosine ¡ The range of cos-1 x is limited differently l Note pg 258 for domain, range of other functions 4

Evaluating Inverse Functions ¡ Consider cos-1(-0. 5) l ¡ We are asking what angle has a cosine value of -0. 5 Cosine negative in quadrants 2 and 3 l 2 But for cos-1(x) we look only in 1 & 2 Calculator also capable of evaluating inverse trig functions -1 5

Try It Out ¡ Consider these Note: newer calculators will have these functions – find in Catalog 6

Composition of Trig Functions and Inverses ¡ Recall that in general l f-1(f(x)) ¡ For trig functions this is the same l l ¡ = f(f-1(x)) = x sin(arcsin(x)) = arcsin(x)) The restriction on the domain and range of the inverse functions must apply Thus l l sin-1(sin(3)) could not be 3 Note calculator results 7

Composition of Trig Functions and Inverses ¡ Try these … with and without calculator 8

Solving Inverse Trig Equations ¡ Given ¡ Strategy l Isolate the sin-1 x l Take the sine of both sides of the equation 9

Try it Out ¡ Try this one 10

Assignment Lesson 3. 5 ¡ Page 265 ¡ Exercises 1 – 65 EOO ¡ 11