Pythagorean Theorem is derived from the right angle triangle. This theorem is very important and useful for all calculus part and geometrical part. In this article we discuss about Pythagorean identities. It is also known as Pythagorean trigonometric identities mean expressing the Pythagorean Theorem in terms of trigonometric functions. It also includes formula of sum of angles and it shows the basic relations between sine and cosine functions and from this basis other identities are derived. For deriving the identities we also have to know the theorem and the theorem states that in a right angle triangle the square of hypotenuse is equal to sum of square of base and square of height.
There are 3 Pythagorean identities. We discuss about all three identities in theory and mathematical forms. Suppose we have given a unit circle, we mark a point on the circle the point is vertex of the right angle and it is represented by the coordinates. Suppose the coordinates are (sinθ, cosθ).the given circle is unit circle so hypotenuse is 1. And the two legs of right angle triangle in a unit circle are sinθ and cosθ. By using Pythagorean Theorem we can write as (sin^2θ+cos^2θ=1). This basic equation is known as first Pythagorean identity.
For the second Pythagorean identity we start from the first Pythagorean identity. First write the identity (sin^2θ+cos^2θ=1). Now we divide each term by (cos^2θ). While dividing we have to remember that (sinθ/ cosθ= tanθ) and (1/ cosθ= secθ). After dividing we get (tan^2θ+1=sec^2θ). This equation is known as second Pythagorean identity.
Now we derive the third Pythagorean identity. For this again we have to use the first Pythagorean identity. First we write the equation (sin^2θ+cos^2θ=1). Now we divide each term by (sin^2θ). While dividing the equation we have to remember some trigonometric formula such as (cosθ/ sinθ= cotθ) and (1/ sinθ= cosecθ). After dividing each term we get (1+ cot^2θ=cosec^2θ). This equation is the third Pythagorean identity.
Pythagorean identities list are...
1. (sin^2θ+cos^2θ=1)
2. (tan^2θ+1=sec^2θ)
3. (1+cot^2θ=cosec^2θ)
Some Pythagorean identities problems. First problem is, suppose we have given secx= (-2/3) and tanx>0 then we have to find values of other trigonometric functions. Solution of this problem is first find (cosx=1/secx=-3/2), then by using first identity find (sinx). After this we can easily determined the remaining functions.
Second problem is suppose we have to solve (sinθcos^2θ-sinθ). A Solution of this problem is that first take out the common term and then we use the first Pythagorean identity. Finally we get the result as (-sin^3θ).
