Trig Identities Calculator
MathVerify the Pythagorean trigonometric identities for a given angle.
sin^2(theta) + cos^2(theta)
Always equals 1, the fundamental Pythagorean identity.
1 + tan^2(theta)
1.567844
Always equals sec^2(theta).
1 + cot^2(theta)
2.761048
Always equals csc^2(theta).
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Frequently Asked Questions
What are the Pythagorean identities?
Three related identities, all derived from the Pythagorean theorem applied to the unit circle: sin^2(theta) + cos^2(theta) = 1, 1 + tan^2(theta) = sec^2(theta), and 1 + cot^2(theta) = csc^2(theta).
Why does sin^2(theta) + cos^2(theta) always equal 1?
On the unit circle, sine and cosine are the y and x coordinates of a point exactly 1 unit from the origin, so by the Pythagorean theorem, x^2 + y^2 = 1^2 for every angle.
How are the other two identities derived from the first?
Dividing every term of sin^2(theta) + cos^2(theta) = 1 by cos^2(theta) gives 1 + tan^2(theta) = sec^2(theta). Dividing instead by sin^2(theta) gives 1 + cot^2(theta) = csc^2(theta).
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