Hard

The Pythagorean Identity and Reciprocal Relations

Simple Explanation

The six trigonometric ratios are not independent — they are linked by fixed relationships. The most important is the Pythagorean identity, sin²θ + cos²θ = 1, which holds for every angle θ. Also, tanθ always equals sinθ/cosθ.

Why Do We Need It?

These identities let you find any trigonometric ratio of an angle from just one known ratio, without needing to know the angle itself or measure a triangle.

See It

sinθ and cosθ as the legs of a right triangle with hypotenuse r

Formula

The Pythagorean Trigonometric Identity

sin²θ + cos²θ = 1 also: tanθ = sinθ/cosθ, cotθ = cosθ/sinθ

The squares of sine and cosine of the same angle always add up to exactly 1 — a direct consequence of the Pythagorean theorem — and tangent is always the ratio of sine to cosine.

θ: — any angle

When to use it: Whenever you know one trigonometric ratio of an angle and need another, without knowing the angle itself.

Worked Example

Find cosθ given sinθ

If sinθ = 3/5 and θ is acute, find cosθ.

Why Does This Work?

Stated simply: if a radius r reaches a point P, its horizontal projection has length r·cosθ and its vertical projection has length r·sinθ, forming a right triangle with hypotenuse r. The Pythagorean theorem gives (r·cosθ)² + (r·sinθ)² = r² — dividing both sides by r² leaves exactly cos²θ + sin²θ = 1.

Real-Life Example

Checking a GPS position calculation

A navigation system computes a position's sine and cosine components from an angle and needs to verify the calculation is internally consistent.

Since sin²θ + cos²θ must always equal exactly 1, this identity is often used as a quick sanity check that a computed angle's trigonometric values are correct.

Practice

If cosθ = 12/13 and θ is acute, find sinθ (as a fraction's numerator over 13 — give just the numerator).

Hard

Common mistake

Forgetting that sin²θ means (sinθ)², not sin(θ²) — the square applies to the whole sine value, not to the angle before taking the sine.

Quick Review

sin²θ + cos²θ = 1, for every angle θ.
tanθ = sinθ/cosθ.
Proof idea: the Pythagorean theorem applied to the horizontal and vertical projections of a radius.