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7.4 Inverse Trigonometric Functions I OBJECTIVE 1: Understanding and Finding the Exact and Approximate Values of the Inverse Sine Function y sin x y sin 1 x Start with the restricted y sin x, 2 x 2 sine function and reverse the coordinates of each ordered pair If 0 x 1, then the value of sin 1 x is an angle whose terminal side lies in to produce the graph of the inverse sine function. Quadrant I. If 1 x 0, then the value of sin 1 x is an angle whose terminal side lies in Quadrant IV. Definition y sin 1 x Inverse Sine Function The inverse sine function, denoted as y sin 1 x , is the inverse of y sin x , x . 2 2 The domain of y sin 1 x is 1 x 1 and the range is 2 y 2 . (Note that an alternative notation for sin 1 x is arcsin x .) Do not confuse the notation sin 1 x with sin x 1 exponent! Thus, sin 1 x 1 csc x . The negative 1 is not an sin x 1 . sin x 1 To determine the exact value of sin 1 x , think of the expression sin 1 x as the angle on the interval 2 , 2 whose sine is equal to x. Use conventional angle notation, , to represent the value of sin 1 x . If sin 1 x , then . Therefore, the terminal side of angle must lie in 2 2 Quadrant I, in Quadrant IV, on the positive x-axis, on the positive y-axis , or on the negative yaxis. If sin 1 x , then 2 2 and the terminal side of angle lies in Quadrant I, in Quadrant IV, on the positive x axis, on the positive y axis, or on the negative y axis. Steps for Determining the Exact Value of sin1 x Step 1. If x is in the interval 1,1 , then the value of sin 1 x must be an angle in the interval 2 , 2 . Step 2. Let sin 1 x such that sin x . Step 3. If sin 0 , then 0 and the terminal side of angle lies on the positive x-axis. If sin 0 , then 0 and the terminal side of angle 2 lies in Quadrant I or on the positive y-axis. 0 and the terminal side of angle 2 lies in Quadrant IV or on the negative y-axis. If sin 0 , then Step 4. Use your knowledge of the two special right triangles, the graphs of the trigonometric functions, Table 1 from Section 6.4, or Table 2 from Section 6.5 to determine the angle in the correct quadrant whose sine is x. 2 OBJECTIVE 2: Understanding and Finding the Exact and Approximate Values of the Inverse Cosine Function Start with the restricted cosine function and reverse the coordinates of each ordered pair y cos x, 0 x to produce the graph of the inverse y cos 1 x cosine function. If -1 x 0, then the value of cos1 x is an angle whose terminal side lies in Quadrant II. If 0 x 1, then the value of cos 1 x is an angle whose terminal side lies in Quadrant I. Definition Inverse Cosine Function The inverse cosine function, denoted as y cos 1 x , is the inverse of y cos x , 0 x . y cos 1 x The domain of y cos 1 x is 1 x 1 and the range is 0 y . (Note that an alternative notation for cos1 x is arccos x .) 3 Think of the expression cos1 x as the angle on the interval 0, whose cosine is equal to x. It is important to notice that the interval 0, is quite different from the interval used to describe the range of the inverse sine function. It is customary to use the notation, , to represent the value of cos1 x . Therefore, if cos1 x , then 0 . Thus, the terminal side of angle must lie in Quadrant I, in Quadrant II, on the positive y-axis, on the positive x-axis, or on the negative x-axis. If cos 1 x , then 0 and the terminal side of angle lies in Quadrant I, in Quadrant II, on the positive y axis, on the positive x axis, or on the negative x axis. Steps for Determining the Exact Value of cos1 x Step 1. If x is in the interval 1,1 , then the value of cos1 x must be an angle in the interval 0, . Step 2. Let cos1 x such that cos x . and the terminal side of angle 2 lies on the positive y-axis. Step 3. If cos 0 , then If cos 0 , then 0 and the terminal side of angle 2 lies in Quadrant I or on the positive x-axis. and the terminal side of angle 2 lies in Quadrant II or on the negative x-axis. If cos 0 , then Step 4. Use your knowledge of the two special right triangles, the graphs of the trigonometric functions, Table 1 from Section 6.4, or Table 2 from Section 6.5 to determine the angle in the correct quadrant whose sine is x. 4 OBJECTIVE 3: Understanding and Finding the Exact and Approximate Values of the Inverse Tangent Function y tan x y tan x, 2 x 2 Start with the restricted tangent function and reverse the coordinates of each ordered pair to produce the graph of the inverse tangent function. 1 y tan x If x 0, then the value of tan 1 x is an angle whose terminal side lies in Quadrant I. If x 0, then the value of tan 1 x is an angle whose terminal side lies in Quadrant IV. Definition Inverse Tangent Function y tan 1 x The inverse tangent function, denoted as y tan 1 x , is the inverse of y tan x , x . 2 2 The domain of y tan 1 x is all real numbers and the range is y . 2 2 (Note that an alternative notation for tan 1 x is arctan x .) 5 Think of the expression tan 1 x as the angle on the interval , whose tangent is equal to x. Use 2 2 conventional angle notation, , to represent the value of tan 1 x . If tan 1 x , then . 2 2 Therefore, the terminal side of angle must lie in Quadrant I, in Quadrant IV, or on the positive x-axis. If tan 1 x , then 2 2 and the terminal side of angle lies in Quadrant I, in Quadrant IV, or on the positive x axis. Steps for Determining the Exact Value of tan1 x Step 1. The value of tan 1 x must be an angle in the interval 2 , 2 . Step 2. Let tan 1 x such that tan x . Step 3. If tan 0 , then 0 and the terminal side of angle lies on the positive x-axis. If tan 0 , then 0 2 and the terminal side of angle lies in Quadrant I. If tan 0 , then 2 0 and the terminal side of angle lies in Quadrant IV. Step 4. Use your knowledge of the two special right triangles, the graphs of the trigonometric functions, Table 1 from Section 6.4, or Table 2 from Section 6.5 to determine the angle in the correct quadrant whose sine is x. 6 7