7 5 RIGHT TRIANGLES INVERSE TRIGONOMETRIC FUNCTIONS Functions

7. 5 RIGHT TRIANGLES: INVERSE TRIGONOMETRIC FUNCTIONS Functions Modeling Change: A Preparation for Calculus, 4 th Edition, 2011, Connally

The Inverse Sine Function For 0 ≤ x ≤ 1: arcsin x = sin− 1 x = The angle in a right triangle whose sine is x. Example 2 Use the inverse sine function to find the angles in the figure. φ 5 3 θ 4 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, 4 th Edition, 2011, Connally

The Inverse Tangent Function arctan x = tan− 1 x = 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◦ 5. 8 ft using a calculator. θ 100 ft A road rising at a grade of 5. 8% (not to scale) Functions Modeling Change: A Preparation for Calculus, 4 th Edition, 2011, Connally

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. Functions Modeling Change: A Preparation for Calculus, 4 th Edition, 2011, Connally