Pythagorean Identities
sin²θ + cos²θ = 1 for every angle.
About this topic
The unit circle gives the fundamental identity sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = csc²θ. These hold for every angle.
Use them to find one trig function from another, simplify expressions and prove other identities. If sin θ = 3/5 and θ is in quadrant II, then cos²θ = 1 − 9/25 = 16/25, so cos θ = −4/5, with the negative sign chosen because cosine is negative in quadrant II.
The identity is simply the Pythagorean theorem applied to the right triangle formed inside the unit circle, with legs cos θ and sin θ and hypotenuse 1. In calculus it is used to simplify integrals, and in physics to relate the components of a vector or the energy in an oscillator.
Tips: when proving identities, work on one side only and convert everything to sine and cosine; rearranged forms such as sin²θ = 1 − cos²θ are especially useful. A classic mistake is writing sin²θ as sin θ², or forgetting the plus or minus when taking a square root.
Key ideas
- Comes from the unit circle
- 1 + tan²θ = sec²θ
- Used to simplify expressions
Quick check
Use cases: where Pythagorean Identities helps
- Homework: review the key idea behind Pythagorean Identities before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Pythagorean Identities without notes.
- Visual learning: play with the interactive lab to see how Pythagorean Identities changes when you adjust its inputs.
- Teaching a friend: explain Pythagorean Identities in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Pythagorean Identities?
sin²θ + cos²θ = 1 for every angle.
What will I learn in Pythagorean Identities?
Comes from the unit circle. 1 + tan²θ = sec²θ. Used to simplify expressions.
Is Pythagorean Identities part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Pythagorean Identities lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Pythagorean Identities?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
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