Final answer:
The Pythagorean identities, based on the Pythagorean Theorem, are three trigonometric equations that describe relationships between the sine, cosine, tangent, secant, cosecant, and cotangent functions, using the unit circle approach.
Step-by-step explanation:
The Pythagorean identities are derived from the Pythagorean Theorem, which relates the lengths of the sides of a right-angled triangle: a² + b² = c², where 'a' and 'b' are the lengths of the legs and 'c' is the length of the hypotenuse. In terms of trigonometry and the unit circle, the identities are as follows:
- sin²(θ) + cos²(θ) = 1
- 1 + tan²(θ) = sec²(θ)
- 1 + cot²(θ) = csc²(θ)
These identities are useful for solving problems involving trigonometric equations and can be remembered by the unit circle approach, where the hypotenuse is always 1 and the legs are represented by the cosine and sine of the angle θ.