Pythagorean Identity

For any angle \(\theta\), the point on the unit circle has coordinates \((\cos\theta, \sin\theta)\). Since every point on the unit circle is exactly 1 unit from the origin, the coordinates must satisfy the distance formula \(x^2+y^2=1\), which becomes \(\cos^2\theta+\sin^2\theta=1\). This identity is true for every angle, not just special angles, and it links the horizontal and vertical parts of the same unit-circle point.

Use this identity when you know one trig function and need the other, especially in algebraic simplification or solving trig equations. It is also helpful for checking whether trig values are reasonable and for converting between sine and cosine in unit-circle problems.

• Forgetting that the identity uses squares, so writing \(\sin\theta+\cos\theta=1\)
• Choosing the wrong sign when taking the square root after solving for \(\sin^2\theta\) or \(\cos^2\theta\)
• Using the identity without checking the quadrant, which is needed to decide whether the final sine or cosine value is positive or negative