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Unknown Angles in Right Angle Triangles - SOH CAH TOA

(Trigonometric Ratios)

In this section we learn how to use SOH CAH TOA to find unknown angles in right angle triangles.

What You'll find here:

  • We start this section by watching a tutorial
  • We then write a three step method for finding angles, that will always work (do make a note of it).
  • Practice exercises, that can be downloaded as a .pdf worksheet.

Tutorial: Unknown Angles Using SOH CAH TOA

In the following tutorial we learn how to find unknown angles in right angle triangles, using the trigonometric ratios and SOH CAH TOA.

Method

Given a right angle triangle, the method for finding an unknown angle \(a\), can be summarized in three steps:

  • Step 1: Label the side lengths, relative to the angle we're after, using "A", "O" and "H".
  • Step 2: Using the labels, made in step 1, look for the only one of the words "SOH", "CAH", or "TOA" that contains both of the letters "O" and "H", or "A" and "H", or "O" and "A". Write the corresponding trigonometric ratio for the unknown angle; \(sin(a) = \frac{O}{H}\), \(cos(a) = \frac{A}{H}\), or \(tan(a) = \frac{O}{A}\).
  • Step 3: replace the letters, "O" and "H", or "A" and "H", or "O" and "A", by their actual values and find the angle using the correct inverse trigonometric function.

Exercise 1

In each of the following right angle triangles, find the unknown side length marked x:

Note: this exercise can be downloaded as a worksheet to practice with: Worksheet 1

Solution Without Working

  1. \(x = 36.9^{\circ}\)

  1. \(x = 26.6^{\circ}\)

  1. \(x = 69.4^{\circ}\)

  1. \(x = 60.0^{\circ}\)

  1. \(x = 46.6^{\circ}\)

  1. \(x = 34.2^{\circ}\)

  1. \(x = 63.5^{\circ}\)

  1. \(x = 32.0^{\circ}\)


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