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1. 7.5 RIGHT TRIANGLES: INVERSE TRIGONOMETRIC FUNCTIONS Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

2. The Inverse Sine Function For 0 ≤x≤ 1: arcsinx = sin−1x = The angle in a right triangle whose sine is x. 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◦ φ 5 3 θ 4 Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

3. The Inverse Tangent Function arctanx = tan−1x = The angle in a right triangle whose tan is x. 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. 5.8 ft θ 100 ft A road rising at a grade of 5.8% (not to scale) Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

4. Summary of Inverse Trigonometric Functions We define: • the arc sine or inverse sine function as arcsinx = sin−1x = The angle in a right triangle whose sine is x • the arc cosine or inverse cosine function as arccosx = cos−1x = The angle in a right triangle whose cosine is x • the arc tangent or inverse tangent function as arctanx = tan−1x = 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−1x cosθ = x means θ = cos−1x tan θ = x means θ = tan−1x. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally