How to Find Cosine from Sine of an Angle?

Last Updated : 14 Feb, 2024

Answer: To find the cosine of an angle from the sine of that angle, use the relationship $\cos(\theta) = \sqrt{1 - \sin^2(\theta)}$ in a right-angled triangle or apply the cofunction identity $(\cos(\theta) = \sin(90^\circ - \theta))$ 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:

$\cos(\theta) = \sqrt{1 - \sin^2(\theta)}$

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 $\cos(\theta) = \sqrt{1 - \sin^2(\theta)}$ 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 $\cos(\theta) = \sin(90^\circ - \theta)$ if the angle is complementary to another known angle. This can simplify calculations in some cases.

Example:

If $\sin(\theta) = \frac{3}{5}$ , then

$\cos(\theta) = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}$

Alternatively, using the cofunction identity:

$\cos(\theta) = \sin\left(90^\circ - \sin^{-1}\left(\frac{3}{5}\right)\right)$

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.

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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.