The Pythagorean identities connect trigonometry to the geometry of right triangles and circles. They are called Pythagorean because they come directly from a right triangle with legs sin θ and cos θ on the unit circle. These identities matter because they let you rewrite trig expressions, simplify equations, and solve for unknown trig values.
They are some of the most useful tools in algebra, precalculus, calculus, and physics.
Key Facts
- sin^2 θ + cos^2 θ = 1
- 1 + tan^2 θ = sec^2 θ
- 1 + cot^2 θ = csc^2 θ
- On the unit circle, the point at angle θ is (cos θ, sin θ).
- The identity sin^2 θ + cos^2 θ = 1 comes from x^2 + y^2 = 1 with x = cos θ and y = sin θ.
- If sin θ = a, then cos^2 θ = 1 - a^2, but the sign of cos θ depends on the quadrant.
Vocabulary
- Pythagorean identity
- A trigonometric identity derived from the Pythagorean theorem, such as sin^2 θ + cos^2 θ = 1.
- Unit circle
- A circle with radius 1 centered at the origin of a coordinate plane.
- Sine
- For an angle θ on the unit circle, sin θ is the y-coordinate of the point on the circle.
- Cosine
- For an angle θ on the unit circle, cos θ is the x-coordinate of the point on the circle.
- Identity
- An equation that is true for every value in its domain.
Common Mistakes to Avoid
- Writing sin θ + cos θ = 1 is wrong because the identity uses squares: sin^2 θ + cos^2 θ = 1.
- Taking the square root without considering sign is wrong because cos θ = ±sqrt(1 - sin^2 θ) depends on the quadrant.
- Treating sin^2 θ as sin(2θ) is wrong because sin^2 θ means (sin θ)^2, not the sine of double the angle.
- Using 1 + tan^2 θ = csc^2 θ is wrong because the correct identity is 1 + tan^2 θ = sec^2 θ.
Practice Questions
- 1 If sin θ = 3/5 and θ is in Quadrant I, find cos θ using a Pythagorean identity.
- 2 Simplify the expression 1 - cos^2 θ + tan^2 θ cos^2 θ.
- 3 Explain why sin^2 θ + cos^2 θ = 1 is true for every angle θ on the unit circle.