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5.6.1 5.6: Inverse Trigonometric Functions – Differentiation Because none of the trigonometric functions are one-to-one, none of them have an inverse function. To overcome this problem, the domain of each function is restricted so as to produce a one-to-one function. Inverse sine function: y sin 1 x if and only if sin y x and y , 2 2 Properties of the inverse sine function: sin 1 sin x x for 2 x 2 sin sin 1 x x for 1 x 1 Example 1: 3 1 1 Evaluate sin 1 and sin . 2 2 5.6.2 Example 2: 3 Evaluate cot sin 1 7 Example 3: Evaluate sin sin 1 0.54 . Example 4: Evaluate sin sin 1 2 . Example 5: Evaluate sin 1 sin 4 Example 6: 5 Evaluate sin 1 sin 6 5.6.3 Inverse cosine function: y cos 1 x if and only if Properties of the inverse cosine function: cos 1 cos x x for 0 x cos cos 1 x x for 1 x 1 Example 7: 1 Evaluate cos 1 2 Example 8: 5 Evaluate cos 1 cos 6 cos y x and y 0, 5.6.4 Inverse tangent function: y tan 1 x lim tan 1 x x if and only if 2 tan y x and y , 2 2 and lim tan 1 x x 2 Graph of y tan 1 x : Inverse secant, cosecant, and cotangent functions: These are not used as often, and are not defined consistently. Our book defines them as follows: y cot 1 x if and only if cot y x and y 0, y csc 1 x if and only if csc y x and y , 0 0, 2 2 y sec 1 x if and only if sec y x and y 0, , 2 2 An important identity: cos 1 x sin 1 x 2 5.6.5 Differentiation of the inverse sine function: Since y sin x is continuous and differentiable, so is y sin 1 x . We want to find its derivative. d 1 sin 1 x dx 1 x2 Example 9: Differentiate f x sin 1 2 x 7 . , 1 x 1 5.6.6 Differentiation of the inverse cosine function: d 1 cos 1 x dx 1 x2 , 1 x 1 Differentiation of the inverse cosine function: d 1 tan 1 x dx 1 x2 Derivatives of other inverse trigonometric functions: d 1 sin 1 x dx 1 x2 d csc1 x dx x 1 x2 1 d 1 cos 1 x dx 1 x2 d 1 sec 1 x dx x x2 1 d 1 tan 1 x dx 1 x2 d 1 cot 1 x dx 1 x2 5.6.7 Example 10: Find the derivative of f x e tan 1 2x . Example 11: Find the derivative of y csc 1 tan x . Example 12: Find the derivative of f ( x) x3 arccos 2 x . Example 13: Find the equation of the line tangent to the graph of f ( x) arctan x at the point where x 1 .