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7.5 RIGHT TRIANGLES: INVERSE TRIGONOMETRIC FUNCTIONS Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

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The Inverse Sine Function Example 2 Use the inverse sine function to find the angles in the figure. Solution Using our calculator’s inverse sine function: sin θ = 3/5 = 0.6 so θ = sin −1 (0.6) = 36.87 ◦ sin φ = 4/5 = 0.8 so φ = sin −1 (0.8) = 53.13 ◦ Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally For 0 ≤ x ≤ 1: arcsin x = sin −1 x = The angle in a right triangle whose sine is x. θ 3 4 5 φ

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The Inverse Tangent Function Example 3 The grade of a road is 5.8%. What angle does the road make with the horizontal? Solution Since the grade is 5.8%, the road climbs 5.8 feet for 100 feet; see the figure. We see that tan θ = 5.8/100 = 0.058. So θ = tan −1 (0.058) = 3.319 ◦ using a calculator. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally arctan x = tan −1 x = The angle in a right triangle whose tan is x. θ 5.8 ft 100 ft A road rising at a grade of 5.8% (not to scale)

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Summary of Inverse Trigonometric Functions We define: the arc sine or inverse sine function as arcsin x = sin −1 x = The angle in a right triangle whose sine is x the arc cosine or inverse cosine function as arccos x = cos −1 x = The angle in a right triangle whose cosine is x the arc tangent or inverse tangent function as arctan x = tan −1 x = The angle in a right triangle whose tangent is x. This means that for an angle θ in a right triangle (other than the right angle), sin θ = x means θ = sin −1 x cos θ = x means θ = cos −1 x tan θ = x means θ = tan −1 x. We define: the arc sine or inverse sine function as arcsin x = sin −1 x = The angle in a right triangle whose sine is x the arc cosine or inverse cosine function as arccos x = cos −1 x = The angle in a right triangle whose cosine is x the arc tangent or inverse tangent function as arctan x = tan −1 x = The angle in a right triangle whose tangent is x. This means that for an angle θ in a right triangle (other than the right angle), sin θ = x means θ = sin −1 x cos θ = x means θ = cos −1 x tan θ = x means θ = tan −1 x. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

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