How to Find the Angles of a Triangle With Equations

By Kylene Arnold

Often, when mathematicians and scientists use triangles, they are not drawing two-dimensional shapes that can be measured with a protractor. Instead, they construct theoretical triangles that must be solved using equations. There are many of these trigonometric equations, each appropriate in a different circumstance. Sine, cosine and tangent functions allow you to find the missing angles of a right triangle. The Laws of Sines and Cosines allow you to calculate the angles of any other triangle.

Sine Function

Substitute the value of the side opposite the unknown angle and the length of the hypotenuse into the sine function, sin(x) = opposite/hypotenuse.

For example, sin(A) = 9/10.

Complete the division on the right side of the equation.

Sin(A) = 0.9

Take the inverse sine, or arcsine, of both sides. The answer is the measurement of the angle.

Angle A equals approximately 64 degrees.

Cosine Function

Substitute the value of the side adjacent to the unknown angle and the length of the hypotenuse into the cosine function, cos(x) = adjacent/hypotenuse.

For example, cos(A) = 3/4.

Complete the division on the right side of the equation.

Cos(A) = 0.75.

Take the inverse cosine, or arccosine, of both sides. The answer is the measurement of the angle.

Angle A equals approximately 41 degrees.

Tangent Function

Substitute the value of the side opposite the unknown angle and the length of the side adjacent to it into the tangent function, tan(x) = opposite/adjacent.

For example, tan(B) = 2/5.

Complete the division on the right side of the equation.

Tan(B) = 0.4.

Take the inverse tangent, or arctangent, of both sides. The answer is the measurement of the angle.

Angle B equals approximately 22 degrees.

Law of Sines

Substitute the values of two sides and one angle into the Law of Sines equation, sin(A)/a = sin(B)/b = sin(C)/c. You can select any two of the three ratios.

For example, if a = 5 cm, b = 6 cm and A = 50 degrees, the equation would become sin(50)/5 = sin(B)/6.

Multiply both sides by the denominator of the fraction with the unknown angle.

For instance, 6[sin(50)/5] = sin(B).

Simplify the expression.

Sin(B) equals approximately 0.92.

Take the inverse sine of both sides to solve for the missing angle.

Angle B is approximately 67 degrees.

Examine the values of the other angles to determine if the angle seems reasonable. If the angles do not add up to 180 degrees, this is a reference angle, not a final answer.

For instance, if the other angles measure 56 and 57 degrees, 67 degrees is reasonable, since 56 + 57 + 67 = 180. But if the other angles are 33 and 34 degrees, then the triangle needs a larger angle to bring the total to 180.

Subtract the reference angle from 180 to find the measurement of the angle.

180 - 67 = 113; therefore, angle B is approximately 113 degrees.

Law of Cosines

Substitute the lengths of the triangle's sides into the Law of Cosines equation, c^2 = a^2 + b^2 - 2ab(cos(C)).

For instance, if a = 2 cm, b = 4 cm and c = 4 cm, then the equation would become 4^2 = 2^2 + 4^2 - 2(2)(4)(cos(C)).

Simplify the expression.

The equation simplifies to 16 = 4 + 16 - 16cos(C) or 16 = 20 - 16cos(C).

Rearrange the equation algebraically to isolate cos(C).

-0.25 = cos(C).

Take the inverse cosine of both sides of the equation to solve for C.

Angle C equals approximately 104.5 degrees.

About the Author

Kylene Arnold is a freelance writer who has written for a variety of print and online publications. She has acted as a copywriter and screenplay consultant for Advent Film Group and as a promotional writer for Cinnamom Bakery. She holds a Bachelor of Science in cinema and video production from Bob Jones University.