How to Find the Cosine of Common Angles?

Here are some common ways to find the cosine of some common angles:

Understand the Common Angles

Memorize or familiarize yourself with the cosine values for some common angles, such as 0°, 30°, 45°, 60°, and 90°. Knowing these values can help you find the cosine of other angles by using trigonometric relationships.

Apply Trigonometric Identities

Use trigonometric identities and relationships to find cosine values for other angles. For example:

Use the fact that cosine is an even function, meaning that cos⁡(−θ) = cos⁡(θ). So, if you know the cosine of a positive angle, you also know the cosine of its negative counterpart.

Use the periodicity property of cosine, which means that cos⁡(θ) = cos⁡(θ ± 360°) or cos⁡(θ) = cos⁡(θ ± 2π). This property allows you to find equivalent angles within one period.

Using Trigonometric Functions and Quadrant

To apply trigonometric functions and the quadrant rule, we need to understand how trigonometric functions behave in each quadrant and how to use reference angles to find the values of trigonometric functions for angles outside of the primary range (0 to 90 degrees or 0 to π/2 radians). Let’s go through the steps:

Quadrant Rule for cosine functions

• In the first quadrant (0 to 90 degrees or 0 to π/2 radians), all trigonometric functions (sine, cosine, tangent) are positive.
• In the second quadrant (90 to 180 degrees or π/2 to π radians), cosine function is negative.
• In the third quadrant (180 to 270 degrees or π to 3π/2 radians), cosine function is negative.
• In the fourth quadrant (270 to 360 degrees or 3π/2 to 2π radians), only the cosine function is positive.

Use of Reference Angles

For angles outside of the primary range, we can use reference angles within the primary range to find trigonometric function values.

• If an angle θ is given, we can find its reference angle α by subtracting the nearest multiple of 90 degrees or π radians from θ.
• Once we have the reference angle α, we use the quadrant rule to determine the signs of trigonometric functions.

Using Right Angle Triangle

If you have a right-angled triangle, apply the cosine function using the identified angle:

[Tex]\cos(\theta) = \frac{\text{length of adjacent side }}{\text{length of hypotenuse}} [/Tex]

Finding the cosine of common angles is a fundamental aspect of trigonometry, it is important for solving various mathematical and real-world problems involving angles and triangles. Common angles such as 0°, 30°, 45°, 60°, and 90° have specific cosine values that are frequently used in calculations.

In this article we will learn different methods to find the cosine of some common angles.