Cosθ

Definition

The term cosθ represents the cosine of an angle θ in a right triangle, which is the ratio of the length of the adjacent side to the length of the hypotenuse. This trigonometric function is essential in understanding relationships between angles and sides in right triangles, as well as in the unit circle where it describes the x-coordinate of a point corresponding to that angle.

5 Must Know Facts For Your Next Test

1. Cosine values range from -1 to 1, with cos(0°) = 1 and cos(90°) = 0.
2. In the unit circle, cosθ gives the x-coordinate of a point on the circumference corresponding to angle θ.
3. The cosine function is periodic with a period of 360°, meaning that cos(θ + 360°n) = cos(θ) for any integer n.
4. Cosine is an even function, which means that cos(-θ) = cos(θ).
5. The values of cosine for common angles like 0°, 30°, 45°, 60°, and 90° are often memorized: cos(0°) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, and cos(90°) = 0.

Review Questions

• How can you use cosθ to find unknown lengths in a right triangle?
  • To find unknown lengths in a right triangle using cosθ, you can apply the definition of cosine as the ratio of the length of the adjacent side to the length of the hypotenuse. If you know either two sides or one side and one angle, you can rearrange this relationship. For example, if you have the length of the hypotenuse and want to find the adjacent side, you would multiply cosθ by that hypotenuse length.
• Discuss how cosθ relates to other trigonometric functions like sine and tangent.
  • Cosθ is closely related to sine and tangent through fundamental identities. The relationship between these functions can be observed in their definitions: sinθ = opposite/hypotenuse and tanθ = sinθ/cosθ. This means that knowing cosθ allows you to find sine if you know either side lengths or angles. Additionally, since tanθ is derived from sine and cosine, changes in cosθ will directly affect tanθ, making it essential to understand their interconnections.
• Evaluate how knowledge of cosine values for specific angles enhances your ability to solve real-world problems involving right triangles.
  • Understanding cosine values for specific angles significantly improves problem-solving skills in real-world applications such as engineering, architecture, and physics. For example, when calculating forces acting on an object or determining heights and distances in navigation, knowing exact values like cos(30°) or cos(45°) allows for quicker calculations without a calculator. This foundational knowledge enables you to apply trigonometric principles effectively and efficiently in practical scenarios.

Related terms

Sine (sinθ): The sine of an angle θ, which is the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.

Tangent (tanθ): The tangent of an angle θ, defined as the ratio of the sine and cosine of that angle, or the length of the opposite side divided by the length of the adjacent side.

Unit Circle:

A circle with a radius of one centered at the origin of a coordinate plane, used to define trigonometric functions including cosine and sine for all real numbers.