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Inverse Trigonometric Functions 4.7

Inverse Trigonometric Functions 4.7. y. y = sin x -  / 2 < x <  / 2. 1. -  / 2. x.  / 2. -1. Domain : [ -  / 2 ,  / 2 ] Range : [ -1, 1 ]. The Inverse Sine Function.

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Inverse Trigonometric Functions 4.7

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  1. Inverse Trigonometric Functions4.7

  2. y y = sin x - /2 < x < /2 1 - /2 x /2 -1 Domain: [- /2, /2] Range:[-1, 1] The Inverse Sine Function The inverse sine function, denoted by sin-1, is the inverse of the restricted sine function y = sin x, - /2 <x < / 2. Thus, y = sin-1x means sin y = x, where - /2 <y < /2 and –1 <x< 1. We read y = sin-1x as “ y equals the inverse sine at x.”

  3. 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.

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

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

  6. The inverse cosine function,denoted by cos-1, is the inverse of the restricted cosine function y = cos x, 0<x <. Thus, y = cos-1x means cos y = x, where 0<y <and –1 <x< 1. The Inverse Cosine Function

  7. Text Example Find the exact value of cos-1 (-3 /2)

  8. Text Example Find the exact value of cos-1 (2 /2)

  9. The inverse tangent function, denoted by tan-1, is the inverse of the restricted tangent function y = tan x, -/2 <x </2. Thus, y = tan-1x means tan y = x, where - /2 <y </2 and –  < x < . The Inverse Tangent Function

  10. Text Example Find the exact value of tan-1 (-1)

  11. Text Example Find the exact value of tan-1 (3)

  12. The Sine Function and Its Inverse sin (sin-1x) = x for every x in the interval [-1, 1]. sin-1(sin x) = x for every x in the interval [-/2,/2]. The Cosine Function and Its Inverse cos (cos-1x) = x for every x in the interval [-1, 1]. cos-1(cos x) = x for every x in the interval [0, ]. The Tangent Function and Its Inverse tan (tan-1x) = x for every real number x tan-1(tan x) = x for every x in the interval (-/2,/2). Inverse Properties

  13. Example

  14. Example

  15. Example

  16. Example

  17. Using you Calculator Find the angle in radians to the nearest thousandth. Then find the angle in degree.

  18. Example • The following formula gives the viewing angle θ, in radians, for a camera whose lens is x millimeters wide. Find the viewing angle in radians and degrees for a 28 millimeter lens.

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