Section 4.7 Inverse Trigonometric Functions

A brief review….. 1.If a function is one-to-one, the function has an inverse that is a function. 2.If the graph of a function passes the horizontal line test, then the function is one- to-one. 3.Some functions can be made to pass the horizontal line test by restricting their domains.

More… 4.If (a,b) is a point on the graph of f, then (b,a) is a point on the graph of f-inverse. 5.The domain of f-inverse is the range of f. 6.The range of f-inverse is the domain of f. 7.The graph of f-inverse is a reflection of the graph of f about the line y = x.

-π/2 π/2 y = sin x is graphed below. The restricted portion is highlighted.

The inverse sine function written y = sin −1 x or y = arcsin x The domain of y = sin x is restricted to y = sin -1 x means that sin y = x (inverse x & y swapped) sin -1 x is the angle, between –π/2 and π/2, (inclusive), whose sine value is x.

Find the exact value (in radians) of each of the following: Think: the angle in whose sine is ½. The answer is because it is the angle in whose sine is ½.

4.7 – Inverse Trig Functions y = cos x The inverse cosine function, written y = cos −1 x or y = arccos x, is the angle between 0 and π whose cosine is x. In other words, y = cos −1 (x) if x = cos y and y is in [0,π] [0,π]

The inverse cosine function The domain of y = cos x is restricted to y = cos -1 x means that cos y = x. cos -1 x is the angle, between 0 and π, inclusive, whose cosine value is x.

Find the exact value (in radians) of each of the following:

4.7 – Inverse Trig Functions y = tan x The inverse tangent function, written y = tan −1 x or y = arctan x, is the angle between and whose tangent is x. In other words, y = tan −1 x if x = tan y and < y <

The inverse tangent function The domain of y = tan x is restricted to (doesn’t include –π/2 and π/2, undefined at these) y = tan -1 x means that tan y = x. tan -1 x is the angle, between –π/2 and π/2, whose tangent value is x.

Find the exact value (in radians) of each of the following: tan (1) tan

Evaluating inverse functions For exact values, use your knowledge of the unit circle. For approximate values, use your calculator (be careful to watch your MODE).

Examples

More Examples

Evaluating composite functions Composite functions come in two types: 1.The function is on the “inside”. 2.The inverse is on the “inside”. In either case, work from the “inside out”. Be sure to observe the restricted domains of the functions you are dealing with. Sometimes the function and inverse will “cancel” each other but, again, watch your restricted domains. For values not on the unit circle, draw a sketch and use right triangle trigonometry.

Examples

Weird Examples Use a right triangle to write the following expression as an algebraic expression: