Pythagorean identity (sin² + cos² = 1)
The Pythagorean Identity
The Pythagorean Identity is a topic in Trigonometric Functions in the Common Core State Standards. It is aligned to Standard F-TF.8, which requires students to prove the Pythagorean identity and use it to find sine, cosine, or tangent given one value and the quadrant.
The Pythagorean identity \(\sin^2\theta+\cos^2\theta=1\) comes from the unit circle and finds one ratio from the other by quadrant.
Theory
The Pythagorean identity follows from the right triangle on the unit circle:
It rearranges to \(1-\sin^2\theta=\cos^2\theta\) and \(1-\cos^2\theta=\sin^2\theta\).
The identity:
How to use the identity
- Substitute the known ratio into \(\sin^2\theta+\cos^2\theta=1\).
- Solve for the square of the unknown.
- Take the square root.
- Choose the sign using the quadrant.
Use \(\sin^2\theta+\cos^2\theta=1\).
| \(\cos^2\theta\) | \(=\) | \(1-\dfrac{9}{25}=\dfrac{16}{25}\) |
| \(\cos\theta\) | \(=\) | \(\dfrac45\) |
Solve for \(\sin\theta\); it is positive in quadrant II.
| \(\sin^2\theta\) | \(=\) | \(1-\dfrac14=\dfrac34\) |
| \(\sin\theta\) | \(=\) | \(\dfrac{\sqrt3}{2}\) |
Rearrange the identity.
| \(1-\sin^2\theta\) | \(=\) | \(\cos^2\theta\) |
\(\sin 30^\circ=\dfrac12,\ \cos 30^\circ=\dfrac{\sqrt3}{2}\).
| \(\left(\dfrac12\right)^2+\left(\dfrac{\sqrt3}{2}\right)^2\) | \(=\) | \(\dfrac14+\dfrac34=1\) |
Common pitfalls
Frequently asked questions
What is the Pythagorean identity?
\(\sin^2\theta+\cos^2\theta=1\).
Where does it come from?
The right triangle on the unit circle, with hypotenuse 1.
How do you find cosine from sine?
\(\cos\theta=\pm\sqrt{1-\sin^2\theta}\), sign by quadrant.
Does \(\sin^2\theta\) mean \(\sin(\theta^2)\)?
No — it means \((\sin\theta)^2\).