The cosine of an angle can be calculated from its sine using trigonometric identities and relationships.
Understanding the Relationship
The core relationship stems from the fact that the sine and cosine functions are related by a 90-degree phase shift. This relationship is expressed in the following identity:
- cos(θ) = sin(90° - θ) (where θ is the angle in degrees)
- cos(θ) = sin(π/2 - θ) (where θ is the angle in radians)
This identity tells us that the cosine of an angle is equal to the sine of its complement. A complementary angle is an angle that, when added to the original angle, sums to 90 degrees (or π/2 radians).
Calculation Methods
Here are practical methods to calculate cosine from sine:
1. Using the Complementary Angle Identity
- Find the complement: Subtract the given angle from 90 degrees (or π/2 radians).
- Calculate the sine: Evaluate the sine of the resulting complementary angle. This will equal the cosine of your original angle.
Example:
If you know that sin(30°) = 0.5, you can calculate cos(30°) as follows:
- Complementary angle: 90° - 30° = 60°
- Calculate sine: cos(30°) = sin(60°) which is approximately 0.866
2. Using the Pythagorean Identity
Another common approach involves using the Pythagorean trigonometric identity, which states:
- sin²(θ) + cos²(θ) = 1
If you have sin(θ), follow these steps:
- Square the sine: Calculate sin²(θ).
- Subtract from one: Calculate 1 - sin²(θ).
- Take the square root: cos(θ) = ±√(1 - sin²(θ)). The sign (+ or -) will depend on the quadrant the angle lies within.
Example:
If you know that sin(30°) = 0.5
- Square the sine: sin²(30°) = (0.5)² = 0.25
- Subtract from one: 1 - 0.25 = 0.75
- Take the square root: cos(30°) = √(0.75) = approximately 0.866. Since 30 degrees is in the first quadrant, the cosine is positive.
Table Summary
| Method | Formula | Steps |
|---|---|---|
| Complementary Angle Identity | cos(θ) = sin(90° - θ) or cos(θ) = sin(π/2 - θ) | 1. Find the complement. 2. Calculate the sine of the complement. |
| Pythagorean Identity | cos(θ) = ±√(1 - sin²(θ)) | 1. Square the sine. 2. Subtract from one. 3. Take the square root. Choose correct sign based on the quadrant. |
Important Notes
- Always ensure you are using consistent units (degrees or radians) throughout your calculations.
- When using the Pythagorean identity, pay close attention to the quadrant of the angle to determine if the cosine is positive or negative.
