key term - Cosine

Definition

Cosine is a trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It plays a crucial role in understanding periodic functions, as it describes how angles map onto the unit circle and is essential in graphing waveforms and solving triangles.

5 Must Know Facts For Your Next Test

1. The cosine function is often abbreviated as 'cos' and can be defined using the coordinates of points on the unit circle.
2. The values of cosine for standard angles, such as 0°, 30°, 45°, 60°, and 90°, are commonly memorized: cos(0°) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, and cos(90°) = 0.
3. The graph of the cosine function is a wave that oscillates between -1 and 1, with a period of 360° or 2π radians.
4. Cosine is an even function, meaning that cos(-x) = cos(x), which reflects its symmetry about the y-axis.
5. In the context of triangles, the Law of Cosines relates the lengths of sides to the cosine of one angle, providing a way to find unknown side lengths or angles.

Review Questions

• How does cosine relate to the unit circle and what are its implications for understanding periodic functions?
  • Cosine is defined using the unit circle where for any angle θ, cos(θ) corresponds to the x-coordinate of the point on the circle. This connection allows us to understand how cosine behaves periodically, repeating its values every 360° or 2π radians. This periodic nature makes it essential for modeling wave-like phenomena in various fields such as physics and engineering.
• In what ways can the Law of Cosines be applied in solving triangles, and how does this relate to finding unknown sides or angles?
  • The Law of Cosines states that for any triangle with sides a, b, and c opposite to angles A, B, and C respectively, the formula a² = b² + c² - 2bc * cos(A) can be used. This relationship allows us to find an unknown side when two sides and the included angle are known or to calculate an angle when all three sides are known. Understanding this application directly ties back to how cosine connects side lengths with their respective angles.
• Evaluate how knowledge of trigonometric identities involving cosine can enhance problem-solving strategies in complex equations.
  • Trigonometric identities such as cos²(x) + sin²(x) = 1 or the double angle formula cos(2x) = cos²(x) - sin²(x) allow for manipulation and simplification of complex equations. By leveraging these identities, students can transform difficult expressions into more manageable forms. This skill enhances problem-solving strategies by providing alternative pathways to reach solutions in geometry, calculus, and other mathematical contexts.

Related terms

sine:

Sine is another trigonometric function that represents the ratio of the length of the opposite side to the hypotenuse in a right triangle.

tangent: Tangent is the trigonometric function that represents the ratio of the length of the opposite side to the adjacent side in a right triangle.

unit circle:

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane, used to define trigonometric functions in terms of angles.