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.

- Skill level:
- Moderate

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### Things you need

- Scientific calculator

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## Instructions

- 1
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.

- 2
Apply the following formula:

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

- 3
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 between them

- 1
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.

- 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)).

- 3
Apply the following formula to find angle CAB, if required:

CAB = 180 - ABC - BCA

## Given two sides and an angle not between them

- 1
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.

- 2
Apply the following formula:

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

- 3
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).

## Given three sides

#### Tips and warnings

- 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.