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Trigonometry

Pythagorean identity

For every angle, the squares of its sine and cosine add up to 1, so one gives the other up to a sign.

sin2⁡θ+cos2⁡θ=1
Divided by cos²θtan2⁡θ+1=sec2⁡θ
Divided by sin²θ1+cot2⁡θ=csc2⁡θ

Calculator

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−1.5−1−0.50.511.5−1−0.50.51(cos θ, sin θ)cossin
  • θ = 143° = 2.496 rad (quadrant II)
  • sin θ = 0.6018, cos θ = −0.7986, tan θ = −0.7536
  • sin²θ + cos²θ = 0.3622 + 0.6378 = 1

What the symbols mean

sin⁡θ
the sine of the angle (between −1 and 1)
cos⁡θ
the cosine of the angle (between −1 and 1)
θ
any angle

When to use it

Use it to find cos θ from sin θ (or the reverse) without finding θ itself, to simplify expressions such as 1−sin2⁡x, and to prove other identities. The identity only fixes the size of the answer: choose the sign from the quadrant. Cosine is positive in quadrants I and IV, sine in I and II.

Worked example

sin θ = 0.6 and θ is obtuse. Find cos θ.

  1. Start from the Pythagorean identity.

    sin2⁡θ+cos2⁡θ=1

  2. Subtract sin2⁡θ and take the square root. The sign is set by the quadrant θ is in, which the identity alone cannot tell you.

    cos⁡θ=±1−sin2⁡θ

  3. Positive root (θ in quadrant I or IV):

    cos⁡θ=1−0.62
    =1−0.36
    =0.64
    =0.8

  4. Negative root (θ in quadrant II or III):

    cos⁡θ=−1−0.62
    =−1−0.36
    =−0.64
    =−0.8

Answer cos θ = ±0.8; θ is obtuse (quadrant II), so cos θ = −0.8

Why it works

On the unit circle the point at angle θ is (cos⁡θ,sin⁡θ), and every point on that circle is distance 1 from the centre. Pythagoras on the right triangle under the point gives cos2⁡θ+sin2⁡θ=1.

Common mistakes

  • Forgetting the ± and assuming the positive root.
  • Reading sin2⁡θ as sin⁡(θ2). It means (sin⁡θ)2.
  • Writing sin⁡θ+cos⁡θ=1. Only the squares add to 1.