# Trigonometric Functions / Trigonometric Ratios

In these lessons, we will look at the three basic trigonometric functions (or trigonometric ratios), Sine, Cosine and Tangent and how they can be used to find missing sides and missing angles. We will also learn how to solve multi-step SOHCAHTOA problems.

The following diagram shows how to use SOHCAHTOA. Scroll down the page for examples and solutions. ### Trigonometric Functions: Sine of an Angle

We first consider the sine function.

The sine of an angle is the ratio of the opposite side to the hypotenuse side. Sine is usually abbreviated as sin. Sine θ can be written as sin θ.

You may use want to use some mnemonics to help you remember the trigonometric functions. One common mnemonic is to remember the Indian Chief SOH-CAH-TOA.

Example:
Calculate the value of sin θ in the following triangle. Solution: Using the Sine Formula (the SOH formula)
This is part 1 of a video that teaches how to use the SOH formula in high school Trigonometry.

This is part 2 of a video that teaches how to use the SOH formula in high school Trigonometry.

### Trigonometric Functions: Cosine of an Angle

Next, we consider the cosine function.

The cosine of an angle is the ratio of the adjacent side and hypotenuse side. Cosine is usually abbreviated as cos. Cosine θ can be written as cos θ.

Example:
Calculate the value of cos θ in the following triangle. Solution: Using the Cosine Formula (the CAH formula)
This video illustrates how to use the Cosine formula (the CAH Formula) in high school trigonometry courses.

### Trigonometric Functions: Tangent of an Angle (#tangent}

Next, we first consider the tangent function.

The tangent of an angle is the ratio of the opposite side and adjacent side. Tangent is usually abbreviated as tan. Tangent θ can be written as tan θ.

Example:
Calculate the value of tan θ in the following triangle. Solution: Using the Tangent Formula (the TOA formula)
This video illustrates how to use the Tangent formula (the TOA Formula) in high school trigonometry.

### Solving problems using SOHCAHTOA

Have a look at the following video for more examples of the trigonometric functions.

A review of basic trigonometry: Sine, Cosine & Tangent including how to use the ratios to find missing sides and missing angles on right triangles.

Find the trigonometric ratios for the right triangle

Multiple-Step SOH CAH TOA Problems - Example 1
This video illustrates how to use the SOH, CAH and TOA formulas from basic trigonometry to solve problems that involve the use of more than one formula.

Example:
Two ladders are leaning against a building as shown. What is the angle below the shorter angle?

Multiple-Step SOH CAH TOA Problems - Example 2
A warbler and a cardinal are perched at the top of two trees. Given the angle of inclination between the apple and the birds, which bird is closer to the apple and by how much?

Multiple-Step SOH CAH TOA Problems

Example:
How long is side UX?

Multiple-Step SOH CAH TOA Problems

Example:
What is the measure of the angle?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 