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.