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Inverse Trigonometry (Right Triangles)

Overview

Lesson 41 used an angle to find a ratio; this lesson runs the process backwards. When two sides of a right triangle are known, their ratio feeds an inverse trigonometric function — sin⁻¹, cos⁻¹, or tan⁻¹ — to recover the measure of the angle. You will evaluate inverse functions on the calculator, find indicated angles in triangles, and solve word problems, including a ladder against a wall and the angle a steep street makes with the horizontal.

Vocabulary

Inverse Trigonometric Functions
The functions sin⁻¹, cos⁻¹, and tan⁻¹, which take a trigonometric ratio as input and return the measure of the angle that produces it.
Reference Angle
The angle being referred to in a trigonometric ratio; the right angle is never used as a reference angle.
Opposite Side
The side directly across the triangle from the reference angle.
Adjacent Side
The side next to the reference angle that is not the hypotenuse.
Hypotenuse
The side opposite the right angle — always the longest side of the right triangle.

Main Ideas

  1. Choose the inverse function that matches the two sides you know: opposite and hypotenuse → sin⁻¹; adjacent and hypotenuse → cos⁻¹; opposite and adjacent → tan⁻¹.
  2. The setup order matters: form the ratio from the sides first, then apply the inverse function to that ratio — the inverse function undoes the ratio, not the sides.
  3. An inverse function returns an angle, so the answer is in degrees; round angle measures as directed, usually to the nearest tenth.
  4. The two acute angles of a right triangle are complementary: if one inverse calculation gives 65.1°, the other acute angle is 24.9°.
  5. In word problems, draw the right triangle, mark the known sides relative to the angle you want, and only then choose the function.

Lesson Notes

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Notes page 1

Lesson Video

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Practice Problems

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Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.