The Cosine function (abbreviated as cos) is the perfect partner to the Sine function. Together, they define circular motion, waves, and the fundamental nature of triangles. While Sine is often associated with the "height" of an angle, Cosine is associated with the "width" or horizontal distance.
The name comes from "complementary sine," abbreviated to co-sine, highlighting the deep relationship between the two functions.
1. Cosine in a Right Triangle
In a right-angled triangle, the cosine of an angle θ is defined as the ratio of the adjacent side to the hypotenuse.
[Image of cosine ratio in right triangle]This provides the "C" in the mnemonic SOH CAH TOA.
2. Cosine on the Unit Circle
Just like Sine, Cosine is best understood using the Unit Circle (a circle with a radius of 1). If you draw a line from the origin at angle θ:
[Image of cosine on unit circle]- The x-coordinate of the point on the circle is cos(θ).
- (The y-coordinate is the sine).
This means that at 0°, the point is on the far right of the circle, so the x-coordinate (cosine) is 1. At 90°, the point is at the very top, so the x-coordinate is 0.
3. The Cosine Graph
The graph of the Cosine function looks very similar to the Sine wave, but with one key difference: it starts at the peak (1) rather than at zero.
[Image of cosine wave graph]Key Features:
- Starting Point: At x=0, cos(x)=1.
- Shape: It follows the same wave pattern as sine but is shifted to the left by 90°.
- Range: Like sine, it oscillates between -1 and 1.
4. Common Cosine Values
Understanding these standard values is crucial for geometry and physics.
| Angle (θ) | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| cos(θ) | 1 | ~0.866 | ~0.707 | 0.5 | 0 |
5. The Law of Cosines
The Law of Cosines is a powerful generalization of the Pythagorean Theorem. It allows you to find a missing side of any triangle (not just right-angled ones) if you know two sides and the angle between them.
Notice that if the angle C is 90°, cos(90°) is 0, and the formula simplifies back to c² = a² + b².
Conclusion
Cosine is essential for describing horizontal distance, periodic waves, and solving complex geometric problems. From calculating the force of a vector to designing stable architectural structures, the cosine function is a fundamental tool in the mathematician's kit.