How to find the missing angle of a triangle using trigonometry

Use the Cosine Law.

Let $\triangle ABC$ have sides $a$, $b$, and $c$. We are using the usual convention that the length of the side opposite vertex $A$ is called $a$, and so on.

Let $\theta=\angle C$. Then the Cosine Law says that $$c^2=a^2+b^2-2ab\cos \theta.$$ Since we know $a$, $b$, and $c$, we can use the above formula to calculate $\cos\theta$. Then we can use the $\cos^{-1}$ button on the calculator to find $\theta$ to excellent accuracy.

We can use the Cosine Law three times to get the three angles. But we only need to do the calculation for two of the angles: If we have them, the third can be easily found.

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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Video Transcript

For the given figure, find the measure of angle 𝜃 in degrees to two decimal places.

As the triangle is right-angled, we can use the trigonometrical ratios sine, cosine, and tangent, shortened to sin, cos, and tan. In any right-angled triangle, sin 𝜃 is equal to the opposite divided by the hypotenuse, cos 𝜃 is equal to the adjacent side divided by the hypotenuse, and tan 𝜃 is equal to the opposite divided by the adjacent.

Our first step is to label the triangle. The longest side of a right-angled triangle is the hypotenuse. The other two sides are determined by which angle we’re dealing with: which one is opposite the angle 𝜃 and which side is adjacent to the angle 𝜃. Well, in this case, the side labelled three is adjacent to the angle 𝜃 and the other side is opposite the angle.

As our measurements are on the adjacent and the hypotenuse, we’re going to use the ratio cos 𝜃 equals the adjacent divided by the hypotenuse. Substituting the values into the formula gives us cos 𝜃 equals three divided by eight or three-eighths.

In order to calculate the angle, we need to use the inverse function or cos to the minus one. Therefore, 𝜃 equals inverse cos of three-eighths. Ensuring our calculator is in degree mode, we can type this in, giving us an answer 𝜃 equals 67.98 degrees. This value has been rounded to two decimal places.

The trigonometrical ratios can be used to find missing angles or missing sides in right-angled triangles. Whenever we wish to find a missing angle, we will have to use the inverse function: inverse sin, inverse cos, or inverse tan.