Pythagorean Trigonometric Identities Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
If sin⁡(θ)=35\sin(\theta) = \frac{3}{5} and θ\theta is in Quadrant I, find cos⁡(θ)\cos(\theta) using the Pythagorean identity.

Solution

  1. 1
    Start with the Pythagorean identity: sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1.
  2. 2
    Substitute sin⁡(θ)=35\sin(\theta) = \frac{3}{5}: (35)2+cos⁡2(θ)=1\left(\frac{3}{5}\right)^2 + \cos^2(\theta) = 1, so 925+cos⁡2(θ)=1\frac{9}{25} + \cos^2(\theta) = 1.
  3. 3
    Solve: cos⁡2(θ)=1−925=1625\cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25}, so cos⁡(θ)=±45\cos(\theta) = \pm\frac{4}{5}.
  4. 4
    Since θ\theta is in Quadrant I, cos⁡(θ)>0\cos(\theta) > 0, so cos⁡(θ)=45\cos(\theta) = \frac{4}{5}.

Answer

cos⁡(θ)=45\cos(\theta) = \frac{4}{5}
The Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 directly relates sine and cosine. When you know one and the quadrant, you can find the other. The quadrant determines the sign of the result.

About Pythagorean Trigonometric Identities

The fundamental identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 and its derived forms: 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta and 1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta.

Learn more about Pythagorean Trigonometric Identities →

More Pythagorean Trigonometric Identities Examples

Example 2 medium

Simplify the expression [formula].

Example 3 medium

Prove that [formula].

Example 4 hard

Simplify [formula].

← All Pythagorean Trigonometric Identities Examples Pythagorean Trigonometric Identities Concept