Introduction

Understanding the value of different trigonometric functions, particularly the cosine, is essential in many fields, from engineering to physics. This article will guide you through a series of methods to find the value of cos 220° using trigonometric identities and the unit circle. Additionally, we will introduce a third method using a cubic equation derived from triple angle identities.

Determining the Value of Cos 220° Using Trigonometric Identities

When trying to find the value of cos 220°, it's helpful to use the properties of the unit circle and trigonometric identities. Here’s a step-by-step guide on how to do this:

Step 1: Determine the Reference Angle

The angle 220° is in the third quadrant. To find the reference angle, subtract 180° from 220°. This gives us:

220° - 180° 40°

Step 2: Determine the Sign of the Cosine Function in the Third Quadrant

In the third quadrant, the cosine function is negative. Thus, we have:

cos 220° -cos 40°

Step 3: Calculate the Cosine of the Reference Angle

The exact value of cos 40° is not a commonly known angle, but it can be approximated using a calculator or trigonometric tables. The approximate value of cos 40° is:

cos 40° ≈ 0.7660

Therefore, the value of cos 220° is approximately:

cos 220° ≈ -0.7660

Using the Triple Angle Identity to Find Cosine Values

To find the value of cos 20°, we can utilize the triple angle identity for cosine:

cos 3θ 4 cos^3 θ - 3 cos θ

Substitute θ 20° into the identity:

cos 60° 4 cos^3 20° - 3 cos 20°

1/2 4 cos^3 20° - 3 cos 20°

Bring 1/2 to the right side to form a cubic equation:

4 cos^3 20° - 3 cos 20° - 1/2 0

Using a tool like WolframAlpha, we can solve this cubic equation and find the roots. The three zeros are:

-0.76604 -0.17365 0.93969

Based on these results, and knowing that the angle 20° lies in the first quadrant where cosine is positive, we select the positive value:

cos 20° ≈ 0.9397

Unit Circle Method

The value of cos 20° can be calculated using the unit circle. Construct an angle of 20° with the x-axis, and find the coordinates of the corresponding point. The coordinates are approximately (0.9397, 0.342). The value of cos 20° is the x-coordinate:

cos 20° 0.9397

Conclusion

Through these methods, we can accurately find the cosine values of specific angles. Whether using trigonometric identities, cubic equations, or the unit circle, each approach provides valuable insights into trigonometric calculations.

Additional Resources

WolframAlpha: Making the world’s knowledge computable - A powerful computational knowledge engine that can solve complex equations. Trigonometric Calculator - An online tool for solving trigonometric functions. Unit Circle Chart - A visual aid for understanding the unit circle and trigonometric functions.