Pythagorean Identity
The fundamental trigonometric identity.
Formula
Variables
Example
At θ = 30°: sin²30° + cos²30° = 1/4 + 3/4 = 1. ✓
How it works
This identity says that if you take the sine and cosine of the same angle, square them, and add them, the result is always 1. It comes from the unit circle: a point at angle θ has coordinates (cosb8, sinb8), and because that point lies on a circle of radius 1, its x- and y-coordinates satisfy x b2 + y b2 = 1. Substituting x = cosb8 and y = sinb8 gives the identity. It works for every angle, not just acute ones, because the unit circle defines sine and cosine for all angles.
When to use it
Use this when you need to simplify trigonometric expressions, verify answers, or convert between sine and cosine in geometry and calculus problems. It is especially useful in proving identities, solving equations, and checking whether a trig result is consistent.
Worked example, step by step
At b8 = 60 b0: sin 60 b0 = 3/2 cos 60 b0 = 1/2 Square both values: sin b260 b0 = (3/2) b2 = 3/4 and cos b260 b0 = (1/2) b2 = 1/4 Add them: 3/4 + 1/4 = 1 So, sin b260 b0 + cos b260 b0 = 1 the identity is confirmed.
Common mistakes to avoid
- Forgetting to square both sine and cosine
- Using degrees or radians inconsistently when finding the trig values
- Assuming the identity only works for acute angles
Did You Know?
sin²θ + cos²θ = 1 is literally the Pythagorean theorem applied to the unit circle.
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