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High School Precalculus: Help and Review32 chapters | 297 lessons

Instructor:
*Jennifer Beddoe*

The cosine function (cos) is one of the basic trigonometric functions. This lesson will define the cosine, explain how to use it to solve problems, and give some examples. The lesson ends with a quiz to help you retain what you have learned.

There are six main trigonometric functions:

- Sine (sin)
- Cosine (cos)
- Tangent (tan)
- Secant (sec)
- Cosecant (csc)
- Cotangent (cot)

These functions are used to relate the angles of a triangle with the sides of that triangle. Trigonometric functions are important when studying triangles and modeling periodic phenomena, such as waves, sound, and light.

The cosine function is defined as the ratio of the side of the triangle adjacent to the angle divided by the hypotenuse.

One way to remember this ratio along with the ratios for the other most common trigonometric functions is with the mnemonic SOHCAHTOA:

SOH = **S**ine is **O**pposite over **H**ypotenuse

CAH = **C**osine is **A**djacent over **H**ypotenuse

TOA = **T**angent is **O**pposite over **A**djacent

This ratio can be used to solve problems involving distance or height, or if you need to know an angle measure.

Example:

What is cosΦ?

Cosine is the ratio of the side adjacent to the angle over the hypotenuse.

cosΦ = 8/17

cosΦ = 0.4706

Trigonometric functions are called **periodic functions** because they repeat over a given period.

Look at the graph of the cosine function:

You can see that the graph repeats itself at a distance of 2π. Therefore, we can say that the cosine function has a period of 2π. Usually, when looking at the cosine function in this way, you don't use degree measure, but **radians**. Radian is the standard unit of angle measurement used in mathematics. A full circle is 2π radians, which is equal to 360°.

In trigonometry, a **unit circle** is a circle centered on the origin (0,0) of a coordinate plane with a radius of 1.

Any point on the circle corresponds to the sine and cosine made by the angle created by a line from the center of the circle to that point and the x-axis. If the point is (x, y), the relationship looks like this -

sinθ = y

cosθ = x

For example:

Using the unit circle, determine the value of cos π/3?

Find the point on the circle that corresponds to π/3 (HINT: It's in Quadrant I in the upper right of the graph).

What is the point on the edge of the circle at that point?

It is (1/2, √(-3)/2) Since 1/2 is the x-value, it is the value of cos π/3.

So, the answer to the question is 1/2.

In this way, the unit circle can be used to find the sine or cosine of any angle, even those that are negative or over 360°

The cosine function is one of the main trigonometric functions. It is important for finding distances or height and can also be used to find angle measures. The unit circle is a tool that can be helpful for finding the value of cosine for all types of angles, especially those that might not be readily apparent.

Knowing the above details on cosine can help you to:

- Name the six main trigonometric functions
- State the cosine function
- Calculate the cosine of an angle using the unit circle

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High School Precalculus: Help and Review32 chapters | 297 lessons

- How to Add, Subtract, Multiply and Divide Functions 6:43
- How to Compose Functions 6:52
- Applying Function Operations Practice Problems 5:17
- Compounding Functions and Graphing Functions of Functions 7:47
- Domain & Range of Composite Functions: Definition & Examples 5:58
- Inverse Functions 6:05
- Understanding and Graphing the Inverse Function 7:31
- One-to-One Functions: Definitions and Examples 4:11
- Manipulating Functions and Solving Equations for Different Variables 9:47
- Cosine: Definition & Overview
- Go to Understanding Function Operations: Help and Review

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