The Pythagorean Theorem works on right triangles. If you consider the
http://www.youtube.com/watch?v=OmJ5fxyXrfg James Sousa: Fundamental Identities: Reciprocal, Quotient, Pythagorean
The proof of the Pythagorean identity for sine and cosine is essentially just drawing a right triangle in a unit circle, identifying the cosine as the
Most people rewrite the order of the sine and cosine so that the sine comes first.
The two other Pythagorean identities are:
To derive these two Pythagorean identities, divide the original Pythagorean identity by
Derive the following Pythagorean identity:
Solution: First start with the original Pythagorean identity and then divide through by
Simplify the following expression:
Note that factoring the Pythagorean identity is one of the most powerful applications. This is very common and is a technique that you should feel comfortable using.
Prove the following trigonometric identity:
Solution: Group the terms and apply a different form of the second two Pythagorean identities which are
Concept Problem Revisited
Cofunctions are not always connected directly through a Pythagorean identity.
Visually, the right triangle connecting tangent and secant can also be observed in the unit circle. Most people do not know that tangent is named “tangent” because it refers to the distance of the line tangent from the point on the unit circle to the
The Pythagorean Theorem states that the sum of the squares of the two legs in a right triangle will always be the square of the hypotenuse.
The Pythagorean Identity states that since sine and cosine are equal to two legs in a right triangle with a hypotenuse of 1, then their relationship is that of the Pythagorean Theorem.
1. Derive the following Pythagorean identity:
2. Simplify the following expression.
3. Simplify the following expression.
3. Note that initially, the expression is not the same as the Pythagorean identity.