How to find the exact value of cosine of 3pi/4?

Introduction

The cosine function is an essential trigonometric function that relates the angle measure to the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. Evaluating the exact value of cosine for certain angles can be challenging, but with the right approach, it becomes simpler. In this article, we will explore how to find the exact value of cosine for the angle 3π/4.

Finding the Exact Value

To find the exact value of cosine for the angle 3π/4, we can make use of the unit circle. The unit circle is a circle with a radius of 1 unit and provides a visual representation of the values of sine and cosine for different angles. Here’s how we can determine the exact value of cosine for 3π/4:

1. Draw a unit circle: Begin by drawing a circle with a radius of 1 unit.
2. Determine the reference angle: The reference angle for 3π/4 is π/4. This is the positive acute angle formed between the terminal arm of the angle and the x-axis.
3. Locate the point on the unit circle: Since cosine represents the x-coordinate of a point on the unit circle, we need to locate the point corresponding to the reference angle of π/4.
4. Determine the x-coordinate: The x-coordinate of the point on the unit circle for the angle π/4 is √2/2.
5. Reflect the x-coordinate: Since 3π/4 lies in the second quadrant, we need to take the opposite sign of the x-coordinate from the previous step. Therefore, the x-coordinate for 3π/4 is -√2/2.
6. Hence, the exact value of cosine for 3π/4 is **-√2/2**.

Related FAQs

1. What is the unit circle?

The unit circle is a circle with a radius of 1 unit that is used to determine the values of trigonometric functions for different angles.

2. What is the reference angle?

The reference angle is the positive acute angle formed between the terminal arm of an angle and the x-axis.

3. How can the unit circle help find exact values?

By locating the angle on the unit circle and determining the corresponding coordinates, we can find the exact values of trigonometric functions.

4. What is the connection between cosine and the unit circle?

In the unit circle, the x-coordinate of a point represents the cosine value for a specific angle measure.

5. Which quadrant is 3π/4 located in?

The angle 3π/4 is located in the second quadrant.

6. How does the sign of the x-coordinate change in the second quadrant?

In the second quadrant, the x-coordinate becomes negative.

7. What is the cosine value for π/4?

The cosine value for π/4 is √2/2.

8. What happens to the cosine value for 3π/4 compared to π/4?

When we move from π/4 to 3π/4, the cosine value changes to its opposite, resulting in -√2/2.

9. Can we use a calculator to find the exact value of cosine for 3π/4?

While calculators can approximate the value, to find the exact value, it is best to rely on geometry and the unit circle.

10. How does the exact value of cosine differ from an approximate value?

The exact value represents the precise mathematical value, while an approximate value is a rounded or decimal representation.

11. Why is it important to know the exact values of trigonometric functions?

Exact values are useful in trigonometric equations, geometric calculations, and other mathematical applications.

12. How can I practice finding exact values for other angles?

By studying the unit circle and practicing with different angles, you can improve your ability to find exact values for other trigonometric functions.

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