Pythagorean Identities

Pythagorean identities convert among trig functions, simplify expressions, and solve equations. Their geometry keeps algebraic sign choices grounded.

Intuition and core definition

The fundamental identity $\sin^2\theta+\cos^2\theta=1$ comes from the unit circle equation $x^2+y^2=1$. Dividing by $\cos^2\theta$ gives $1+\tan^2\theta=\sec^2\theta$ where cosine is nonzero; dividing by $\sin^2\theta$ gives $\csc^2\theta=1+\cot^2\theta$ where sine is nonzero.

Notation, language, and conditions

$\sin^2\theta$ means $(\sin\theta)^2$, not $\sin(\theta^2)$. Derived identities have domain restrictions inherited from division. Identities are equations true for every input where both sides are defined, unlike equations solved for selected angles.

Why this idea matters

Pythagorean identities translate the unit circle's distance equation into relationships among trigonometric functions, including their necessary domains.

A dependable method

1. Choose the identity containing the known and requested functions.
2. State quadrant information and relevant domains.
3. Isolate the squared unknown function.
4. Take square roots with a sign chosen from the quadrant.
5. Check the identity and a numerical or unit-circle representation.

Worked example

Representations and interpretation

A radius-one right triangle has legs $\cos\theta$ and $\sin\theta$, so the Pythagorean theorem yields their squared sum. Reciprocal identities correspond to rescaling that triangle by a coordinate.

Reasoning about variations

The identity determines magnitude but not sign when only a squared value is isolated. Without a quadrant or interval, both signs may be possible. Domain restrictions matter when deriving tangent or cotangent identities.

Common mistakes

How to check your work

• Substitute exact values into the original identity.
• Use ASTC/quadrant coordinate signs.
• Verify derived tangent as sine divided by cosine.

Practice

1. If $\cos\theta=3/5$ and $\theta$ is in Quadrant IV, find $\sin\theta$.
2. Complete $1+\tan^2\theta=\ ?$.
3. What condition is needed for the tangent identity?

Answers and brief solutions

Synthesis and transfer

When one coordinate of a unit-circle point is known, the identity supplies the other's magnitude, but the quadrant—not the square root alone—determines its sign.

If a unit-circle point has cosine $-5/13$ and lies in Quadrant II, then sine has magnitude $12/13$ and must be positive. Dividing gives tangent $-12/5$, while reciprocals yield secant $-13/5$ and cosecant $13/12$. The identity supplies magnitude from the circle equation, but quadrant data supplies signs. Deriving $1+\tan^2\theta=\sec^2\theta$ divides by $\cos^2\theta$, so inputs with cosine zero remain outside that identity's domain even though the original sine-cosine identity still holds. Tracking that restriction prevents an algebraically derived statement from being applied at a forbidden angle.

Related topics

• Trigonometric Identities
• Unit Circle
• Reciprocal Trigonometric Functions

Teaching and accessibility note