Explore and apply all three Pythagorean identities. Given one trig value and a quadrant, find all six trig values with exact fractions and full step-by-step derivations. Or verify any identity numerically at any angle.
Input
Enter any angle θ to verify that all three Pythagorean identities hold exactly at that value. Each identity should equal 1.
Results
| Function | Exact Value | Identity used |
|---|
sin²θ + cos²θ = 1
sin²θ = 1 − cos²θ
cos²θ = 1 − sin²θ
1 + tan²θ = sec²θ
tan²θ = sec²θ − 1
sec²θ − tan²θ = 1
1 + cot²θ = csc²θ
cot²θ = csc²θ − 1
csc²θ − cot²θ = 1
x = cos θ, y = sin θ, x² + y² = 1
The unit circle is the circle centered at the origin with radius 1. By definition, any point on the unit circle satisfies x² + y² = 1 (the Pythagorean theorem for a right triangle with hypotenuse 1).
When we define cos θ as the x-coordinate and sin θ as the y-coordinate of the point at angle θ on this circle, substituting gives us immediately:
(cos θ)² + (sin θ)² = 1
This is the fundamental Pythagorean identity — it is true for every angle θ without exception, because every point on the unit circle satisfies x² + y² = 1.
Start: sin²θ + cos²θ = 1
Divide every term by cos²θ (assuming cos θ ≠ 0):
sin²θ/cos²θ + cos²θ/cos²θ = 1/cos²θ
tan²θ + 1 = sec²θ
Divide every term by sin²θ (assuming sin θ ≠ 0):
sin²θ/sin²θ + cos²θ/sin²θ = 1/sin²θ
1 + cot²θ = csc²θ
The three identities are really one identity seen from three perspectives — dividing through by sin², cos², or neither.
Our tutors break down Pythagorean identities step by step, helping you understand the why behind each manipulation and build confidence for exams.