The Pythagorean and Quotient Identities
sin²θ + cos²θ = 1 is Pythagoras on the unit circle, and tan θ = sin θ / cos θ is a ratio of coordinates. Use both to simplify expressions and to prove further identities.
What a learner can do afterwards
- Derive sin²θ + cos²θ = 1 from the radius of the unit circle
- Simplify (1 - cos²θ) / sin θ to a single function
- Prove a given identity by working on one side until it matches the other
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Every point on the unit circle obeys x squared plus y squared equals 1. With x as cosine and y as sine, that reads sin squared plus cos squared equals 1. Pythagoras on radius 1 proves it for every angle.
Tangent is a ratio of coordinates. Tan theta equals sin theta over cos theta. Where cosine is zero the ratio has no value. That single line turns many hard forms into familiar ones.
Simplify 1 minus cos squared theta, all over sin theta. The top matches the identity, so it becomes sin squared theta. One sine cancels with the bottom. The whole form collapses to sin theta.
To verify an identity, work on one side until it matches the other. Never move terms across the equals sign. Start with the messier side and quote each step. Given sin theta is 0.6 acute, cos squared is 1 minus 0.36, so cos theta is 0.8.
Read both identities off the circle, then use them to rewrite one side at a time.
More to read
9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions
Verifying means working on ONE side until it matches the other, never moving terms across the equals sign. Start with the messier side: replace 1 − sin squared with cos squared, or tan with sin over cos, then simplify.
To simplify (1 − cos squared θ) over sin θ, the top becomes sin squared, and one sin cancels to leave sin θ. Each step uses a known identity or plain algebra. Quote the identity beside the step.
Introduction to Trigonometric Identities and Equations
The Pythagorean identity comes from the unit circle itself: x squared + y squared = 1 with x = cos θ and y = sin θ gives sin squared + cos squared = 1. Pythagoras on radius 1 proves it for every angle.
From this one root grow the quotient and reciprocal forms. Divide through by cos squared for the tan secant version, or memorise tan = sin over cos directly.
Introduction to Trigonometric Identities and Equations
Identities are tools for rewriting, not equations to solve: both sides match for every angle. Use them to trade unfamiliar forms for familiar ones before solving or simplifying. Recognition is the skill.
Given sin θ = 0.6 acute, cos θ follows without a calculator: cos squared = 1 − 0.36 = 0.64, so cos θ = 0.8. The identity bridges one function value to another in two lines.
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Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.