Pythagorean Identities

The Pythagorean theorem can be applied to the trigonometric ratios that give rise to the Pythagorean identity. In this step-by-step guide, you will learn the concept of Pythagorean identity.

Tutor-style math help

Pythagorean Identities: what to notice and how to work it

Trigonometry skill

Trigonometry connects an angle to a triangle ratio, a unit-circle coordinate, or a repeating graph. Choosing the right picture makes the problem much easier.

What to notice first

Decide whether the problem is triangle-based, circle-based, or graph-based. Then use the matching definition.

Common student mistake

Do not mix degrees and radians. The angle unit must match the formula, graph scale, or calculator setting.

Key formulas and cues

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)

\(\tan\theta=\frac{\sin\theta}{\cos\theta}\)

\(\sin^2\theta+\cos^2\theta=1\)

A reliable path

Choose the modelUse a right triangle, the unit circle, or a transformed graph.
Track unitsConvert degrees and radians when needed.
Use identitiesReplace complicated trig expressions with equivalent simpler ones.

Worked examples

Right-triangle sine

Example: opposite = 5, hypotenuse = 13

Sine is opposite over hypotenuse.
Substitute 5 and 13.
Leave the ratio simplified.

Answer: \(\sin\theta=\frac5{13}\)

Unit-circle cosine

Example: \(\cos(0)\)

At angle 0, the point is (1, 0).
Cosine is the x-coordinate.
Read the x-value.

Answer: \(1\)

Open pop-up practice

Try one before moving on

Try: In a right triangle, tangent equals which ratio?

Answer: Opposite over adjacent.

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

In mathematics, identity is an equation that holds for all possible values. An equation that contains trigonometric functions and is true for any value that replaces the variable is called trigonometric identity.

A step-by-step guide to Pythagorean identities

Pythagorean identities are important identities in trigonometry derived from the Pythagorean theorem. These identities are used to solve many trigonometric problems in which a trigonometric ratio is given and other ratios are found.

The fundamental Pythagorean identity shows the relationship between \(sin\) and \(cos\), and is the most common Pythagorean identity that says:

\(\color{blue}{sin^2\theta +cos^2\theta =1}\) (which gives the relation between \(sin\) and \(cos\))

There are two other Pythagorean identities as follows:

\(\color{blue}{sec^2\theta -tan^2\theta =1}\) (which gives the relation between \(sec\) and \(tan\))
\(\color{blue}{csc^2\theta -cot^2\theta =1}\) (which gives the relation between \(csc\) and \(cot\))

Pythagorean trig identities

All Pythagorean trig identities are listed below.

\(\color{blue}{sin^2\theta +cos^2\theta =1}\)
\(\color{blue}{1+tan^2\theta =sec^2\theta}\)
\(\color{blue}{1+cot^2\theta =cosec^2\theta}\)

Each of them can be written in different forms with algebraic operations. That is, any Pythagorean identity can be written in three ways as follows:

\(\color{blue}{sin^2θ + cos^2θ = 1 ⇒ 1 – sin^2θ = cos^2 θ ⇒ 1 – cos^2θ = sin^2θ}\)
\(\color{blue}{sec^2θ\ – tan^2θ = 1 ⇒ sec^2θ = 1 + tan^2θ ⇒ sec^2θ – 1 = tan^2θ}\)
\(\color{blue}{csc^2θ\ – cot^2θ = 1 ⇒ csc^2θ = 1 + cot^2θ ⇒ csc^2θ – 1 = cot^2θ}\)

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