How to Use Pythagorean Identities in Trigonometry

Using the Pythagorean theorem, the unit circle, and sin and cosine you can learn how to use these identities to solve and simplify trigonometric equations, make additional trigonometric identities, and evaluate trigonometric functions. Learn how to derive the three Pythagorean identities by manipulating, squaring and transforming them in a few easy steps.

Instructions

1. Sketch a right angle on your piece of paper with a horizontal base. Label the base a, the vertical side b, and label the straight line from a to b, c.

2. Draw a circle, a unit circle, around your right triangle with the base of the triangle on a horizontal line dividing the circle. Divide the circle in half by a vertical line. Position the right triangle with the acute angle of the right triangle pointing to the center of the circle.

3. Relabel the base of the right triangle, a, with cos theta. Label side b, sin theta and relabel line c to equal 1. Your equation for the right triangle is a^2 + b^2 = c^2 or (cos theta)^2 + (sin theta)^2 = 1^2 = 1. This is your first Pythagorean identity.

4. Divide through the last equation with cos^2 theta for the second Pythagorean identity. It reads cos^2 (theta)/cos^2 (theta) + sin^2 (theta)/cos^2 (theta) = 1/cos^2 (theta). This translates to 1 + tan^2 (theta) = sec^2 (theta).

5. Do the same now with sin^2 (theta). You get the equation cos^2 (theta)/sin^2 (theta) + sin^2 (theta)/sin^2 (theta). Translate this equation to cot^2 (theta) + 1 = csc^2 (theta). This is the third Pythagorean identity.