How to Find Cosine from Sine of an Angle?
Answer: To find the cosine of an angle from the sine of that angle, use the relationship in a right-angled triangle or apply the cofunction identity if the angle is complementary to another known angle.
Here is a detailed explanation of how to find the cosine of an angle when you know the sine of that angle:
1. Understand the Relationship:
The relationship between sine and cosine in a right-angled triangle is expressed by the Pythagorean identity:
sin2(θ)+cos2(θ)=1
From this, we can derive the expression for cosine in terms of sine:
2. Identify the Sine Value:
If you know the sine of the angle (sin(θ)), substitute this value into the formula.
3. Calculate the Cosine Value:
Use the formula to calculate the cosine value.
4. Check for Quadrant:
The sign of sin(θ) determines the quadrant of the angle. Adjust the sign of the calculated cosine value accordingly.
5. Alternative Method using Cofunction Identity:
Another approach is to use the cofunction identity if the angle is complementary to another known angle. This can simplify calculations in some cases.
Example:
If , then
Alternatively, using the cofunction identity:
This method allows you to find the cosine of an angle when you are given the sine value, providing an alternative to direct calculation with a calculator.