What Is the Pythagorean Identity (sin²θ + cos²θ = 1)?

The Pythagorean identity is $\sin^2\theta+\cos^2\theta=1$. It is true for every angle $\theta$: the squares of the sine and cosine of the same angle always add to $1$.

Why the identity is true

On the unit circle, the point determined by $\theta$ has coordinates $(x,y)=(\cos\theta,\sin\theta)$. Because the circle has radius $1$, Pythagoras' theorem gives

$$x^2+y^2=1^2.$$

Substituting $x=\cos\theta$ and $y=\sin\theta$ produces

$$\cos^2\theta+\sin^2\theta=1.$$

Reversing the order of addition gives the usual form. This coordinate argument explains why the identity works in every quadrant, including when sine or cosine is negative.

The identity is also a conversion tool: it replaces a squared sine with a cosine expression, or vice versa. This is especially useful when simplifying trigonometric expressions and solving equations that contain both functions, because it can reduce the equation to one trigonometric ratio.

Useful rearrangements

Suppose $\cos\theta=-\frac{3}{5}$ and $\theta$ lies in the second quadrant. Then

$$\sin^2\theta=1-\left(-\frac{3}{5}\right)^2 =1-\frac{9}{25} =\frac{16}{25}.$$

Therefore, $\sin\theta=\pm\frac{4}{5}$. Since sine is positive in the second quadrant, $\sin\theta=\frac{4}{5}$. Geometrically, both signs initially appear because squaring removes sign information; the quadrant restores it.

A quick check uses $\theta=\frac{\pi}{4}$. Since both sine and cosine equal $\frac{\sqrt{2}}{2}$, their squared values are each , so the sum is . This verifies one case, while the unit-circle argument proves the identity for all angles.

Common misconceptions

The statement $\sin\theta+\cos\theta=1$ is false: the identity contains the squares of both ratios. Also, $\sqrt{\sin^2\theta}=|\sin\theta|$, not automatically . Never drop the sign when solving from a squared value; use the stated interval or quadrant to choose the valid sign.

IB exam technique

This is IB Mathematics: Analysis and Approaches SL 3.6: Pythagorean identity and double angles, assessed on Papers 1 and 2. In a response, state the identity, substitute exact values, simplify clearly, and justify the final sign. On Paper 1 especially, retain fractions and surds rather than replacing exact values with calculator decimals.