In mathematics, the study of triangles is called trigonometry. Any unknown values of angles and sides may be discovered using the common trigonometric identities of Sine, Cosine and Tangent. These identities are simple calculations used to convert the ratios of sides into degrees of an angle. Unknown angles are referred to as **angle theta** and may be calculated in various ways, based on known sides and angles.

## Right Triangles

When a triangle contains a 90 degree angle, it is known as a **right angle triangle**, and angle theta can be determined using the acronym **SOHCAHTOA**.

When broken down, this represents that Sine (S) is equal to the length of the side opposite angle theta (O) divided by the length of the hypotenuse (H) so that Sin(X) = Opp/Hyp. Similarly, Cosine (C) is equal to the length of the adjacent side (A) divided by the hypotenuse. (H) Cos(X) = Adj/Hyp. Tangent (T) is equal to the opposite (O) divided by the adjacent (A). Tan(X) = Opp/Adj.

To solve these ratios using a graphing calculator, you use the inverse trig functions -- known as **arcsin**, **arccos** and **arctan** -- and represented on the calculator as SIN^-1, COS^-1, and TAN^-1.

If the length of the opposite side is known as well as the hypotenuse -- corresponding to the SOH in the acronym -- use the arcsin function on the calculator, and then input the two lengths in fractional form.

For example: If the side opposite angle theta has a length of 4 and the hypotenuse has a length of 5, input the ratio into the calculator like this:

SIN^-1(4/5)

This should output a value of approximately 53.13 degrees. If not, make sure the calculator is set to DEGREE mode, and then try again.

## Law of Sines

If no 90 degree angles are present in a triangle, SOHCAHTOA has no meaning in solving for angles. However, if an angle and the length of its opposite side are known, the **Law of Sines** can be used in cooperation with another known side length to find missing angles. The law states that sin A/a = sin B/b = sin C/c.

Broken down, this means that the sine of an angle divided by the length of its opposite side is directly proportional to the sine of another angle divided by the length of its opposite side. To solve, isolate the sine of the unknown angle by multiplying both sides of the equation by the length of angle theta's opposite side.

For example: sin A/a = sin B/b becomes (b * sin A)/a = sin B

In a calculator, given side a = 5, side b = 7, and angle A = 45 degrees, this is seen as SIN^-1((7*SIN(45))/5). This gives angle B a value of approximately 81.87 degrees.

## Law of Cosines

The **Law of Cosines** works on all triangles but is primarily used in instances where the lengths of all sides are known, but none of the angles are known. The formula is similar to the **Pythagoras Theorem** (a^2 + b^2 = c^2) and states c^2 = a^2 + b^2 - 2ab*cos(C). But for purposes of finding theta, it is easier read as cos(C) = (a^2 + b^2 - c^2)/2ab.

For example, if a triangle has three sides measuring 5, 7 and 10, input these values into a graphing calculator as cos^-1((5^2 + 7^2 - 10^2)/(2_5_7)). This calculation outputs a value of approximately 111.80 degrees.

## Practice for Mastery

An important thing to remember is that all triangles are composed of three angles that have a total sum of 180 degrees. Practice the different techniques on different triangles until the process becomes familiar. Sometimes discovering theta is the same as discovering a new way to work around the problem.