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Pythagorean Identity

Understanding the Pythagorean Identity

The Pythagorean Identity is one of the most fundamental relationships in trigonometry. It is directly derived from the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) applied to a right triangle within the unit circle, where the hypotenuse is 11, the opposite side is sin⁡θ\sin \theta, and the adjacent side is cos⁡θ\cos \theta.

The Three Forms of the Identity

The primary Pythagorean Identity is: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

By dividing this fundamental equation by either cos⁡2θ\cos^2 \theta or sin⁡2θ\sin^2 \theta, we can derive two other useful forms:

  1. Divide by cos⁡2θ\cos^2 \theta: tan⁡2θ+1=sec⁡2θ\tan^2 \theta + 1 = \sec^2 \theta

  2. Divide by sin⁡2θ\sin^2 \theta: 1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta

These identities are essential tools for finding missing trigonometric values and simplifying complex expressions.

Finding Trigonometric Values

You can use the Pythagorean Identity to find an unknown trigonometric value if you know another one and the quadrant where the angle lies.

Example: If sin⁡θ=35\sin \theta = \frac{3}{5} and θ\theta is in Quadrant II, find cos⁡θ\cos \theta and tan⁡θ\tan \theta.

Solution:

  1. Start with the main identity: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  2. Substitute the known value: (35)2+cos⁡2θ=1\left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1 925+cos⁡2θ=1\frac{9}{25} + \cos^2 \theta = 1
  3. Solve for cos⁡2θ\cos^2 \theta: cos⁡2θ=1−925=1625\cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}
  4. Take the square root. Since θ\theta is in Quadrant II, the cosine value must be negative: cos⁡θ=−45\cos \theta = -\frac{4}{5}
  5. Find tan⁡θ\tan \theta using the quotient identity (tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}): tan⁡θ=35−45=−34\tan \theta = \frac{\frac{3}{5}}{-\frac{4}{5}} = -\frac{3}{4}

Simplifying Trigonometric Expressions

The identities are also used to rewrite and simplify expressions.

Example: Simplify 1−cos⁡2θsin⁡θ\frac{1 - \cos^2 \theta}{\sin \theta}.

Solution:

  1. Recognize that from sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, we can rearrange it to get: 1−cos⁡2θ=sin⁡2θ1 - \cos^2 \theta = \sin^2 \theta
  2. Substitute this back into the numerator of the original expression: sin⁡2θsin⁡θ\frac{\sin^2 \theta}{\sin \theta}
  3. Cancel out a common factor of sin⁡θ\sin \theta: sin⁡θ\sin \theta

The simplified expression is simply sin⁡θ\sin \theta.