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How to use trigonometry to find an angle

Updated February 21, 2017

Trigonometry is, broadly speaking, a branch of mathematics that concerns itself with the study of triangles and the relationship between their sides and angles. Given some information about a triangle with vertices A, B and C, sides AB, BC and CA and angles ABC, BCA, CAB, you can use a trigonometry formula to calculate a specific angle in all but one case. Which formula you use, however, depends on the kind of information you are given.

Given two sides and an angle between them

Assume you are given sides AB and AC and angle CAB, and need to find angle ABC. The process remains exactly the same if you need to find angle BCA.

Apply the following formula:

x = AC * sin(CAB) / sqrt(AB^2 + AC^2 - (2 * AB * AC * cos(CAB)))

Compare AC^2 with ((2 * AB^2) + AC^2 - (2 * AB * AC * cos(CAB))).

If the two numbers are equal, angle ABC = arcsin(x) = 90. If AC^2 is smaller, angle ABC = arcsin(x). If AC^2 is greater, angle ABC = 180 - arcsin(x).

Given two sides and an angle not between them

Assume you are given sides AB and AC and angle BCA. Both angle ABC and angle CBA will be found in the course of the calculation; if you only require angle ABC, you can stop after completing Step 2.

Apply the following formula to find angle ABC:

x = AC * sin(BCA) / AB

If x > 1, the problem has no solutions. If x = 1, angle ABC is 90. If x < 1, the problem has two solutions. Angle ABC is equal to either arcsin(x) or (180 - arcsin(x)).

Apply the following formula to find angle CAB, if required:

CAB = 180 - ABC - BCA

Given three sides

Assume you are given sides AB, BC and CA, and need to find angle CAB. The process is identical for any of the three angles.

Apply the following formula:

x = (AB^ + AC^2 - BC^2) / (2 * AB * AC)

Check the value of x.

If x > 1 or x < -1, the problem has no solutions. If x = 1 or x = -1, all three angles are 0 and the triangle is, in fact, a line segment. If -1 < x < 1, then angle CAB = arccos(x).

Tip

If you are given two out of three angles in a triangle, the third can be found without using trigonometry, by subtracting the sum of those two angles from 180. Arcsin is marked "sin^-1" and arccos "cos^-1" on most scientific calculators.

Things You'll Need

  • Scientific calculator
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About the Author

Laurel Storm has been writing since 2001, and helping people with technology for far longer than that. Some of her articles have been published in "Messaggero dei Ragazzi", an Italian magazine for teenagers. She holds a Master of Arts in writing for television and new media from the University of Turin.