Key takeaway: The Pythagorean identity states that sin²θ + cos²θ = 1 for every angle θ, which lets you find one of sine or cosine from the other up to a sign.
Grade 11 – Algebra 2F-TF.C.8Trigonometry
Definition of Pythagorean Identity
The Pythagorean identity says that the square of the sine plus the square of the cosine of any angle is exactly 1. It is the Pythagorean theorem wearing trig clothing.
On a circle of radius 1, the point at angle \(\theta\) is \((\cos\theta,\sin\theta)\). Its horizontal and vertical distances from the center are the legs of a right triangle, and the radius is the hypotenuse. Squaring and adding the legs must give \(1^2\). Because the statement is true for all angles, it is called an identity rather than an equation to solve.
Formula and notation
The identity and two rearrangements:
\[\sin^2\theta+\cos^2\theta=1,\quad \sin\theta=\pm\sqrt{1-\cos^2\theta},\quad \cos\theta=\pm\sqrt{1-\sin^2\theta}.\]
The notation \(\sin^2\theta\) means \((\sin\theta)^2\). The \(\pm\) is settled by the quadrant.
Key facts
- It holds in radians and degrees, for positive, negative and large angles.
- Dividing every term by cos²θ gives the companion identity 1 + tan²θ = sec²θ.
- Neither sine nor cosine can exceed 1 in size, a direct consequence.
- The quadrant of θ decides which sign to keep after a square root.
Pythagorean Identity example
An angle \(\theta\) in quadrant II has \(\sin\theta=\frac{8}{17}\). Find \(\cos\theta\) and \(\tan\theta\).
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Rearrange the identity and square: 1 minus \(\frac{64}{289}\) leaves \(\frac{225}{289}\) for the squared cosine.
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The root of that fraction is 15 over 17, with a possible minus sign.
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Quadrant II has negative cosine, so \(\cos\theta=-\frac{15}{17}\).
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Divide sine by cosine: \(\tan\theta=-\frac{8}{15}\).
\(\cos\theta=\)\(-\frac{15}{17}\) and \(\tan\theta=-\frac{8}{15}\).
Common mistake
Watch out
Marcus recalls the Pythagorean identity as \(\sin\theta+\cos\theta=1\). With \(\sin\theta=\frac{8}{17}\) he gets \(\cos\theta=\frac{9}{17}\).
Why it is wrong
The squares matter. Without them the statement is false: at 45° the sum of sine and cosine is about 1.41. Square first, subtract from 1, take a root and pick the sign by quadrant.
Related terms and lessons
- Unit Circle
- Trigonometric Identities
- Reference Angle
- Double-Angle Formulas
- Sum and Difference Formulas
- Algebra 2 · Radians and the Unit CircleLessonWorksheetQuiz
- Precalculus · Trigonometric IdentitiesLessonWorksheetQuiz
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