Pythagorean Identity
The key identity from the unit circle.
Key idea
The key identity from the unit circle.
In depth
sin²θ + cos²θ = 1. It is the equation x² + y² = 1 of the unit circle, with x = cos θ and y = sin θ. It lets you find one function from the other, up to sign.
Worked example
If sin θ = 3/5 in Q1, cos θ = √(1 − 9/25) = 4/5.
Formula
Real-world uses
- Checking calculator results and simplifying expressions in physics formulas
Common mistake
Forgetting the sign when taking the square root.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Pythagorean Identity?
The key identity from the unit circle. It is a topic at the advanced level in Trigonometry.
What is the formula for Pythagorean Identity?
The key formula is sin²θ + cos²θ = 1.
Where is Pythagorean Identity used in real life?
Real-world examples: Checking calculator results and simplifying expressions in physics formulas.
What should I learn before Pythagorean Identity?
Make sure you are comfortable with Graphs of Reciprocal Trig Functions, the previous topic in Trigonometry.
What is a common mistake with Pythagorean Identity?
Forgetting the sign when taking the square root.
How can I practise Pythagorean Identity?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Pythagorean Identity?
Move on to Other Pythagorean Identities next.
Related advanced topics in Trigonometry
- Graph of Tangent – Periodic with vertical asymptotes at odd multiples of π/2.
- Phase Shift – Horizontal translation of a wave.
- Vertical Shift and Midline – Moves the wave up or down.
- General Sinusoid – Combine amplitude, period, phase and shift.
- Graphs of Reciprocal Trig Functions – Secant, cosecant and cotangent graphs.
- Other Pythagorean Identities – Divide the identity by sin² or cos².