Inverse Trigonometric Functions Section 4.5 Inverse Trigonometric Functions

Objectives: 1. To define inverse trigonometric functions. 2. To evaluate inverse trigonometric 3. To graph inverse trigonometric

Remember, to find the rule for an inverse function, you interchange the x and y and solve for y.

In the case of y = sin x, solving x = sin y for y requires a symbol for the inverse. Mathematicians use the notation y = sin-1 x or y = arcsin x. This notation means “y is the angle whose sine is x.”

The notation sin-1 indicates an angle!

Consider the sin function Consider the sin function. Since the function is not one-to-one the inverse is not a function.  2 - -2 1 -1

To make the original function one-to-one we restrict the domain of sin to [ , ] - 2  2  2 - -2 1 -1

This is the graph of f(x) = Sin x.  2 - -2 1 -1

By reflecting Sin x across the line y = x we get the graph of f(x) = Sin-1 x.  2 - -2 1 -1

EXAMPLE 1 y = Sin-1 1. Find y. y = Sin-1 1 is an equivalent expression to sin y = 1. In other words, we want to know the angle whose sin is 1. Since the “s” in sin is capitalized we want the angle from the restricted domain [ , ].  2 -

EXAMPLE 1 y = Sin-1 1. Find y. y = Sin-1 1 y =  2

EXAMPLE 2 Find sin(Sin-1 ) sin(Sin-1 ) = sin = 1 2  6

EXAMPLE 3 Find sin(Cos-1 ) 4 Cos-1 3/4 represents an angle in [0, ]. Since 3/4 is positive it is a first quadrant angle. Therefore you have the following right triangle. x  3 4

EXAMPLE 3 Find sin(Cos-1 ) 4 Use the Pythagorean theorem to find the missing side. x  3 4 32 + x2 = 42 9 + x2 = 16 x2 = 7 x = 7

EXAMPLE 3 Find sin(Cos-1 ) 4  4 7 3 ∴ sin(Cos-1 ) = 3 4 7

EXAMPLE 4 Find Cos-1(- ) 2 Since the cosine is negative, the angle is in the second quadrant. The cos = . The angle in the second quadrant with a reference angle of is the angle  - = . 2  4 3

Homework pp. 196-197

►A. Exercises Graph. 1. y = cos x 1 

►A. Exercises Graph. 2. y = Cos x, x  [0, ] 1 

►A. Exercises Graph. 3. y = Cos-1 x  1 1 

►A. Exercises Graph. 4. y = tan x 1 

►A. Exercises Graph. 5. y = Tan x, x  ( , ) - 2  1 

►A. Exercises Graph. 6. y = Tan-1 x

►A. Exercises Without using a calculator, find the following values. 3 2

►A. Exercises Without using a calculator, find the following values. 13. tan(Sin-1 ) 1 2

►A. Exercises Without using a calculator, find the following values. 15. cos(Sin-1 ) - 5 3

►A. Exercises Use a calculator to determine the following values. 17. Sin-1 0.3420

►B. Exercises Graph the given function over its appropriate restricted domain. (State the restricted domain.) Graph its inverse function on the same set of axes. 21. g(x) = Csc x

►B. Exercises Use the definitions and a calculator to evaluate the following. 23. Cot-1 0.684

►B. Exercises Use the definitions and a calculator to evaluate the following. 27. Sin-1 0.7854

■ Cumulative Review 35. Give the angle of inclination of the line 3x + 4y = 7 to the nearest degree.

■ Cumulative Review 36. Change f(x) = x – to general form. 5 7 1 4

■ Cumulative Review 37. Give the function rule for the line passing through the points (-4, 5), (3, 8.5), and (8, 11).

■ Cumulative Review 38. Find an equivalent expression for f(x) = sec ( – x).  2

■ Cumulative Review 39. Find the inverse of the function f(x) = x – 5. 2 3

y = sin x 1 -1 - -2 2  y = csc x

y = cos x 1 -1 - -2 2  y = sec x

y = tan x 1 -1 - -2 2  y = cot x