Using Trig Ratios to Solve Triangles: Angles
The steps are the same as the ones we use to solve for a side, but the process will look a little different:
1. Choose which trig ratio to use.
- Choose sin, cos, or tan
2. Substitute
- Write the trig ratio and substitute in the values
3. Solve
- Solve for the angle using the inverse ratios.
The inverse ratios start with the ratio and then find the angle that produces this ratio. They are sometimes also called the arcsin, arccos, and arctan. On your calculator, these buttons probably appear as follows:
Example:If we know just two sides of a triangle, we can find the measure of the angles.
1) Find the measure of angle A.
Step 1: Choose which trig ratio to use.
Step 2: Substitute
write our trig ratio:
Then, we substitute in the sides:
Step 3: Solve
To solve for the angle, we must use
the inverse tangent or arctan.
And the tan-1 button on your calculator
to find the answer. Round to the
nearest tenth.
2) Find the measure of angle B.We need to determine how the two sides we know the length of are related to angle A. The 4 in. side is adjacent to 
A and the 7 in. side is opposite A. The ratio that uses adjacent and opposite is the tangent.
A and the 7 in. side is opposite A. The ratio that uses adjacent and opposite is the tangent.Step 2: Substitute
write our trig ratio:
Then, we substitute in the sides:
Step 3: Solve
To solve for the angle, we must use
the inverse tangent or arctan.
And the tan-1 button on your calculator
to find the answer. Round to the
nearest tenth.
We could simply use the fact that the angles of a triangle add up to 180°
to find the last angle. That would give us an answer of 29.7°
. But once again, that assumes we found angle A correctly. The best thing to do will be to work through the steps for angle B. If we get 29.7°
, then we will be fairly confident that we found both the correct answers.
Step 1: Choose which trig ratio to use
Step 2: Substitute
Write our trig ratio:
Then, we substitute in the sides:
Step 3: Solve
To solve for the angle, we must use
the inverse tangent or arctan.
And the tan-1 button on your calculator
to find the answer. Round to the
nearest tenth.
Remember, if your calculator doesn't seem to be giving you the right answer, read your manual or ask someone for help. Each calculator is different, and you have to enter the information in the correct order to get the right answer. Also, make sure your calculator is in Degree mode (not Radians)Step 1: Choose which trig ratio to use
The 4 in. side is opposite B and the 7 in. side is adjacent toB. The ratio that uses adjacent and opposite is once again the tangent.
B and the 7 in. side is adjacent toB. The ratio that uses adjacent and opposite is once again the tangent.Step 2: Substitute
Write our trig ratio:
Then, we substitute in the sides:
Step 3: Solve
To solve for the angle, we must use
the inverse tangent or arctan.
And the tan-1 button on your calculator
to find the answer. Round to the
nearest tenth.
Practice: Find the measure of the angles of these triangles:
1) m
A 2) mB 3) mE 4) mF
Answers: 1) 60° 2) 30° 3) 56.3° 4) 33.7°
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