When I look at which posts people are viewing on this blog, inevitably one of the most popular posts is the one about Right Triangle Trigonometry (SOH-CAH-TOA) and its follow-up post with more practice problems. Today, we’ll expand on that topic by looking at how you can determine the measure of an angle in a right triangle.

Consider this problem:

In the right triangle ABC below, what is the measure, to the nearest degree, of Angle A?

A. 38

B. 44

C. 46

D. 49

E. 53

So how do we find that angle? Well, we know that the side *adjacent * to the angle is 10 units long, and the *hypotenuse* is 14 inches long. Therefore, we should be able to set up an equation involving cosine (the CAH in SOH-CAH-TOA).

That equation would be cos *A* = 10/14. So how do we solve this to find the measure of angle *A*? I’ll give you the procedure for a TI-84 graphing calculator, but you’d do something similar with any other graphing or scientific calculator.

- Go to
**MODE**and make sure you are set to**DEGREE** - Press
**2nd**and then**COS**… it should say cos^{-1} - Enter (10/14) after the cos
^{-1}so that is says cos^{-1}(10/14) and hit**ENTER**

The angle measure rounds to 44 degrees, so the correct answer is Choice B.

Here are a couple of problems that you can try. The solutions appear after the second problem.

1. A ramp is 25 feet long and rises 3 feet vertically as shown in the diagram below. What angle does the ramp make with the ground? Round your answer to the nearest tenth of a degree.

A. 4.1

B. 5.3

C. 6.6

D. 6.9

E. 7.6

This second one is a bit trickier!

2. In the diagram below, *AD* = 13, *CD* = 5, and *BC* = 8 as shown. What is the measure, to the nearest tenth of a degree, of ∠*ABC*?

F. 33.7

G. 41.8

H. 48.1

J. 52.9

K. 56.3

Solutions:

1. D

From the diagram, you can see that the side *opposite* the angle is 3 feet long and the *hypotenuse* is 25 feet long. That means that you can write and solve an equation involving sine (the SOH in SOH-CAH-TOA).

sin θ = 3/25

θ = sin^{-1} (3/25)

θ ≈ 6.9 degrees

2. K

You can start by using Pythagorean Theorem to find the length of segment *AC*.

a² + b² = c²

5² + b² = 13²

25 + b² = 169

b² = 144

b = 12

Let’s put that length into our diagram.

Now you can see that for ∠ *ABC*, the side *opposite* the angle has a length of 12 units and the side *adjacent* to the angle has a length of 8 units. You should be able to write an equation involving tangent (the TOA in SOH-CAH-TOA).

tan θ = 12/8

θ = tan^{-1} (12/8)

θ ≈ 56.3 degrees

If you have any questions about these problems, right triangle trigonometry in general, or anything to do with the ACT, send us an email at info@cardinalec.com