Inverse Trigonometric Functions: Apply It

  • Understand the domain restrictions on inverse sine, cosine, and tangent
  • Find the exact value of expressions involving the inverse sine, cosine, and tangent functions.
  • Use a calculator to evaluate inverse trigonometric functions.
  • Use inverse trigonometric functions to solve right triangles.
  • Find exact values of composite functions with inverse trigonometric functions.

Angles and Surveying

Inverse trigonometric functions are essential tools in navigation and surveying, where angles must be determined from measured distances. Surveyors measure side lengths and use inverse trig functions to find the angles needed to complete their maps and calculations.

Each inverse trigonometric function has a restricted range to ensure it returns only one angle value:

  • [latex]\sin^{-1}(x)[/latex]: domain [−1, 1], range [latex]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right][/latex]
  • [latex]\cos^{-1}(x)[/latex]: domain [−1, 1], range [latex][0, \pi][/latex]
  • [latex]\tan^{-1}(x)[/latex]: domain (−∞, ∞), range [latex]\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)[/latex]

A surveyor stands 50 feet from the base of a cell tower. The top of the tower is 120 feet above eye level. What is the angle of elevation to the top of the tower?

A surveyor measures a horizontal distance of 85 feet from a point to the base of a building. The vertical height to a window is 45 feet. Find the angle of elevation to the window.

A surveyor measures a slope distance of 340 feet with a horizontal distance of 320 feet. What angle does the slope make with the horizontal?

A surveyor calculates that for a particular sight line, [latex]\cos(\theta) = \frac{3}{5}[/latex]. She needs to find [latex]\sin(\theta)[/latex] for additional calculations. Find [latex]\sin\left(\cos^{-1}\left(\frac{3}{5}\right)\right)[/latex].