pythagorean identity (sin and cos)
sin²θ + cos²θ = 1
The Pythagorean identity is a fundamental trigonometric concept that relates the sine and cosine functions of an angle in a right-angled triangle with the hypotenuse of the triangle. It can be stated as:
sin²θ + cos²θ = 1
where θ represents an angle in a right-angled triangle.
This identity can be derived from the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
To see how the Pythagorean identity follows from the Pythagorean theorem, consider a right-angled triangle with one of its angles labeled θ, as shown in the diagram below:
“`
|
| /|
| / |
| / |
______|/___|_____
A B C
“`
In this triangle, we can define the sine and cosine of θ as follows:
sin(θ) = AB / AC
cos(θ) = BC / AC
Using the Pythagorean theorem, we can relate these lengths to the hypotenuse AC:
AB² + BC² = AC²
Substituting the definitions of sine and cosine, we get:
(sinθ)² + (cosθ)² = 1
Hence, the Pythagorean identity is derived.
The Pythagorean identity is useful in many trigonometric applications, including finding missing sides or angles in right-angled triangles, simplifying trigonometric expressions, and proving other trigonometric identities.
More Answers:
Master Even/Odd Identities Of Sine – Simplify Trigonometric Expressions And Prove Identities With EaseThe Pythagorean Identity For Csc And Cot Functions In Trigonometry
The Pythagorean Identity: A Fundamental Trigonometric Concept For Right-Angled Triangles