# Trigonometry Examples

### Trigonometry Examples for the ACT Math Exam

This page has ACT math problems and trigonometry examples for you to study, such as problems on sine, cosine, and tangent.

Sine

Cosine

Tangent

### Trigonometry Examples in Right Triangles

Trigonometry questions on the test will evaluate your understanding of the relationships and functions of sine, cosine, and tangent in right triangles.

Remember the following important trigonometric formulas for calculating the sine, cosine, and tangent of any given angle A, as in the illustration above:

sin A = x/z

cos A = y/z

tan A = x/y

In other words, for any angle, sine is calculated by taking the measurement of the opposite side divided by the measurement of the upper side (hypotenuse) of the triangle.

Cosine is calculated by taking the measurement of the lower side (which is adjacent) divided by the measurement of the upper side (or hypotenuse) of the triangle.

Tangent is calculated by taking the measurement of the opposite side divided by the measurement of the lower or adjacent side of the triangle.

### Trigonometric Relationships

These are the trigonometric relationships for right triangles:

• cos2 A + sin2 A = 1
• cos2 A = 1 − sin2 A
• sin2 A = 1 − cos2 A
• Tangent is always equal to sin ÷ cos.

The trigonometry equations for sine, cosine, and tangent are included above, as well as in our cheat sheet.

### Using Trigonometry on the ACT Math Exam

You will see different types of questions on sine, cosine, and tangent on the test.

Some of these problems will tell you directly that you need to calculate the sine, cosine, or tangent.

Example 1: How is the tangent of x calculated?

Example 2: For any given angle A, sin2 A = ?

However, the majority of questions on the trigonometry part of the math test will only imply indirectly that you need to use one of these  trigonometry equations.

So you will need to understand how to use sine, cosine, or tangent in order to answer the question.

For example, the question might show you a triangle and give you the measurements of the degrees of two of the angles in the triangle, and then ask you to calculate the length of one of the sides of the triangle.

More complex problems will show two triangles embedded inside or partially within each other.

In these cases, you will need to use the trigonometry equations you have learned in order to calculate the degrees or length of the particular part of one of the triangles, and then use that result in another calculation for the second triangle in order to arrive at your final answer.

You will also need to use ACT trigonometry equations in order to understand trigonometric graphing and modeling.

Here are some math equations for trigonometry questions. You can practice these formulas and more in our free sample test.

θ = s ÷ r

θ = the radians of the subtended angle

s = arc length