7-6 The Inverse Trigonometric Functions

1. 7-6 The Inverse Trigonometric Functions Objective: To find values of the inverse trigonometric functions.

2. The Inverse Trigonometric Function When does a function have an inverse? It means that the function is one-to-one. One-to-one means that every x-value is assigned no more than one y-value AND every y-value is assigned no more than one x-value. How do you determine if a function has an inverse? Use the horizontal line test (HLT).

3. y x y = sin x –/2 /2 The Inverse Trigonometric Function Inverse Sine Function Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test. f(x) = sin x does not pass the Horizontal Line Test and must be restricted to find its inverse. sin x has an inverse function on this interval.

4. Angle whose sine is x This is another way to write arcsin x. The Inverse Trigonometric Function The inverse sine function is defined by y = arcsin x if and only if sin y = x. Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians). The domain of y =arcsin x is [–1, 1]. The range of y =arcsin x is [–/2 , /2]. Example 1:

5. The Graph of Inverse Sine

6. Finding Exact Values of sin-1x • Let  = sin-1x. • Rewrite step 1 as sin  = x. • Use the exact values in the table to find the value of  in [-/2 , /2] that satisfies sin  = x.

7. Example • Find the exact value of sin-1(1/2)

8. y x y = cos x  0 The Inverse Trigonometric Function The other inverse trig functions are generated by using similar restrictions on the domain of the trig function. Consider the cosine function: Inverse Cosine Function f(x) = cos x must be restricted to find its inverse. cos x has an inverse function on this interval.

9. Angle whose cosine is x This is another way to write arccos x. The Inverse Trigonometric Function The inverse cosine function is defined by y = arccos x if and only if cos y = x. Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians). The domain of y =arccos x is [–1, 1]. The range of y =arccos x is [0 , ]. Example 2:

10. The Graph of Inverse Cosine

11. p p - < < y 2 2 Inverses of Sine and Cosine Sin(x) Domain: Range: -1≤y≤1 Arccos(x) Domain: -1≤x≤1 Range: 0≤y≤¶ Cos(x) Domain: 0≤x≤¶ Range: -1≤y≤1 Arcsin(x) Domain: -1≤x≤1 Range:

12. The Inverse Trigonometric Function The other trig functions require similar restrictions on their domains in order to generate an inverse. Like the sine function, the domain of the section of the tangent that generates the arctan is

13. Angle whose tangent is x This is another way to write arctan x. The Inverse Trigonometric Function The inverse tangent function is defined by y = arctan x if and only if tan y = x. Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians). The domain of y =arctan x is (-,) . The range of y =arctan x is (–/2 , /2). Example 3:

14. p p - < < x 2 2 Evaluate each expression without using a calculator. If = x, the sinx = -1 and Since sin (-π/2) = -1, then Sin (-1) = (π/2) Whose tangent is √3 Since tan π/3 = √3 then Tan √3 = π/3

15. Find the following: -1 • Find Sin (0.8) with a calculator. Degree mode = 53˚ Radian mode = 0.93 • Find Cos (-0.5) without a calculator. Cos (-0.5) = x means that cos x =-0.5 between 0 and π. Thus, Cos (-0.5) = 2π/3 -1 -1 -1

16. 3 -2 Find with and without a calculator. Hypotenuse² will be (-2)² + 3² = √13 The cos is adj/hyp = 3/√13 Rationalize Denominator = 3√13/13 √13 Calculator answer ≈ 0.83

17. Find the approximate value (calculator) and exact value (without a calculator) csc(cos (-0.4)) -1 5 -2 -0.4 in fraction form is -2/5 Cos = adj/hyp Opp. =√ 5² - (-2)² = √21 Csc = 1/sin = hyp/opp = 5/√21 Rationalize denominator = 5√21/21 Calculator: 1.09

18. Assignment • Page 289 #2, 4, 5 – 8, 11 – 14 • Chapter 7 Test Wednesday