How to Find the Cosine of Common Angles?

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.

The cosine function, denoted as cos(θ), is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.

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

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]

Here are few example for better understanding:

Example 1: Find the cosine of 135 degrees.

Solution:

Use Known Values: We can’t directly find the cosine of 135 degrees from common values, so we’ll use the relationship between trigonometric functions and reference angles.

135 degrees is in the second quadrant, and cosine is negative in the second quadrant.

The reference angle for 135 degrees is 45 degrees.

Calculate: Since the reference angle is 45 degrees and cosine of 45 degrees is [Tex]\frac{1}{\sqrt{2}}[/Tex]​​​, and it is in the second quadrant (where cosine is negative):

Hence, cos⁡(135^∘) = [Tex]-\frac{1}{\sqrt{2}}[/Tex]

Example 2: Find the cosine of 210 degrees.

Solution:

Use Known Values: We can’t directly find the cosine of 210 degrees from common values, so we will use the relationship between trigonometric functions and reference angles.

210 degrees is in the third quadrant, and cosine is negative in the third quadrant.

The reference angle for 210 degrees is 30 degrees.

Calculate: Since the reference angle is 30 degrees and cosine of 30 degrees is [Tex]\frac{\sqrt{3}}{2}[/Tex]​​ and it is in the third quadrant (where cosine is negative):

cos⁡(210^∘) = [Tex]-\frac{\sqrt{3}}{2}[/Tex]

Example 3: In a right triangle, the length of the adjacent side is 3 units and the length of the hypotenuse is 5 units. Find the cosine of the angle θ.

Solution:

Given: Adjacent side (a) = 3 units Hypotenuse (h) = 5 units

Using the formula for cosine:

cos⁡(θ) =[Tex]\frac{adjacent \ side}{Hypotenuse}[/Tex] = [Tex]\frac{3}{5}[/Tex]

So, the cosine of the angle θ[Tex]\frac{3}{5}[/Tex]​.