Pythagorean Identity
In short: The Pythagorean identity states that sin²θ + cos²θ = 1 for every angle θ, so knowing either sine or cosine, plus the quadrant, determines the other value.
Algebra 2TrigonometryF-TF.C.8
What is Pythagorean Identity?
The Pythagorean identity is the Pythagorean theorem wearing trigonometry clothes. A point \((\cos\theta, \sin\theta)\) on the unit circle is exactly one unit from the center, so its coordinates must satisfy \(x^2 + y^2 = 1\).
The result is \(\sin^2\theta + \cos^2\theta = 1\), true for all real \(\theta\). Dividing by \(\cos^2\theta\) gives \(\tan^2\theta + 1 = \sec^2\theta\), and dividing by \(\sin^2\theta\) gives \(1 + \cot^2\theta = \csc^2\theta\). Solving for one value leaves a plus-or-minus sign to settle from the quadrant.
How to spot it
- One trig value is given, such as sin θ = 5/13, and the Pythagorean identity must supply the other values exactly.
- A problem asks to simplify an expression like 1 – cos²θ, or to prove two trig expressions equal.
- The quadrant of the angle is stated, which hints that a sign must be chosen after a square root.
- A right triangle inside a circle of radius 1 has legs called sine and cosine.
The method
- Write the Pythagorean identity and substitute the known value: for example \(\cos^2\theta = 1 – \sin^2\theta\).
- Compute the square and subtract from 1 to get the square of the unknown value.
- Take the square root, keeping both signs for the moment.
- Use the quadrant to choose the sign, then build tangent and other values as quotients.
Pythagorean Identity: solved example
Problem
If \(\sin\theta = \frac{5}{13}\) and \(\theta\) is in Quadrant II, find \(\cos\theta\) and \(\tan\theta\).
Solution
From the identity, \(\cos^2\theta = 1 – \left(\frac{5}{13}\right)^2 = 1 – \frac{25}{169} = \frac{144}{169}\). So \(\cos\theta = \pm\frac{12}{13}\).
Cosine is negative in Quadrant II, so \(\cos\theta = -\frac{12}{13}\), and \(\tan\theta = \frac{5/13}{-12/13} = \)\(-\frac{5}{12}\).
Try it
Question 1
If \(\cos\theta = 0.6\) and \(\theta\) is in Quadrant IV, find \(\sin\theta\).
Show the answer
\(\sin^2\theta = 1 – 0.36 = 0.64\), so \(\sin\theta = \pm 0.8\). In Quadrant IV sine is negative: \(-0.8\).
Question 2
Simplify \(\frac{1 – \sin^2\theta}{\cos\theta}\).
Show the answer
The top equals \(\cos^2\theta\), so the result is \(\cos\theta\).
Pythagorean Identity vs. Pythagorean Theorem
The theorem relates the sides of any right triangle: \(a^2 + b^2 = c^2\), and works for any size. The identity is a trigonometric statement about angles, true for every angle even when no triangle is drawn. It is the theorem applied to a triangle whose hypotenuse is 1.
Keep going
Lessons
Worksheets and quizzes
- Graphs of Trigonometric Functions and Identities – Worksheet
- Graphs of Trigonometric Functions and Identities – Quiz
- Trigonometric Identities – Worksheet
- Trigonometric Identities – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Trigonometry