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Trigonometric Identities

A trig identity is an equation true for every angle. The most important is the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1, which links sine and cosine.

Given one of them and the angle's quadrant, you can solve for the other. The quadrant decides the sign of your answer.

The Pythagorean identity

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 holds for any angle. Rearranged, cos⁡2θ=1−sin⁡2θ\cos^2\theta = 1 - \sin^2\theta, so knowing one gives the square of the other.

Take the square root to find the value, but remember the ±\pm — the quadrant tells you which sign to keep.

Using the quadrant for the sign

Sine is positive in quadrants I and II; cosine is positive in I and IV. So an angle between 90∘90^\circ and 180∘180^\circ (quadrant II) has negative cosine.

In the diagram, angle θ\theta lands in quadrant II. The terminal side reaches up and to the left, so its height, sin⁡θ\sin\theta, is positive while its horizontal reach, cos⁡θ\cos\theta, points left of the yy-axis and is negative.

Compute the magnitude with the identity, then attach the sign the quadrant requires.

sin⁡θ\sin\theta
cos⁡θ\cos\theta
θ\theta
OO
PP

Worked examples

Example 1: cosine from sine

If sin⁡θ=35\sin\theta = \dfrac{3}{5} and 90∘<θ<180∘90^\circ < \theta < 180^\circ, find cos⁡θ\cos\theta.

Use the identitycos⁡2θ=1−(35)2=1625\cos^2\theta = 1 - \left(\tfrac{3}{5}\right)^2 = \tfrac{16}{25}
Take the rootcos⁡θ=±45\cos\theta = \pm\tfrac{4}{5}
Quadrant II: cosine negative−45-\tfrac{4}{5}

Answer: −45-\dfrac{4}{5}

Example 2: sign in quadrant IV

In quadrant IV, is cosine positive or negative?

Cosine positive in I and IVpositive\text{positive}

Answer: Positive

Example 3: sine from cosine

If cos⁡θ=−817\cos\theta = -\dfrac{8}{17} and 180∘<θ<270∘180^\circ < \theta < 270^\circ, find sin⁡θ\sin\theta.

Use the identitysin⁡2θ=1−(−817)2=225289\sin^2\theta = 1 - \left(-\tfrac{8}{17}\right)^2 = \tfrac{225}{289}
Take the rootsin⁡θ=±1517\sin\theta = \pm\tfrac{15}{17}
Quadrant III: sine negative−1517-\tfrac{15}{17}

Answer: −1517-\dfrac{15}{17}

Try one yourself

Common questions

What is the Pythagorean identity?

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1, true for every angle. It lets you find sine from cosine or vice versa.

Why do I need the quadrant?

Taking the square root gives ±\pm. The quadrant tells you which sign is correct for that angle.

Where is cosine negative?

In quadrants II and III, where the x-coordinate is negative.

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