Three Pythagorean Identities in Trigonometry

In math, an identity is an equation which remains always true. This can be trivially true like x = x or usefully true such as the Pythagorean Theorems a² + b² = c²   for right angled triangles.  There are loads of trigonometric identities.  Trigonometric Identities are equations that are true for any right angled triangles.  The Pythagorean Identities in trigonometry are defined as three different identities that   come from the Pythagorean Theorem. The Pythagorean Theorem states that the hypotenuse squared of a right angled triangle is equal to the sum of the square of each   of the other two sides or a squared plus b squared. In Pythagorean Theorem, c stands for the hypotenuse and a   and b stands for the other two sides of the right angled triangle. Fromm this theorem the three identities can be determined from substituting in sine and cosine.

The first Pythagorean identity in Trigonometry:

The first identity probably looks like the Pythagorean theorem.

This identity of Pythagorean Theorem comes from the unit circle. All the right angled triangles formed by this unit circle will have their hypotenuse as 1. With the hypotenuse of 1 and an angle at the origin of the coordinate plane on which the circle is drawn, the relationship between the sides and the sine and cosines can be formed.  This identity is useful when a problem is seen with a sine squared plus a cosine squared. One can use this identity and replace the sine squared plus the cos squared with the 1.

The Second Pythagorean Identity in Trigonometry:

The second identity that comes from the first has one of the sides equal to 1. In order to arrive at this identity, the first identity is divided by the sine squared so as to get the other side equal to 1. I t is known that cosine squared divided by sine squared is cotangent squared and also 1 divided by the sine squared is cosecant squared.

The Third Pythagorean Identity in Trigonometry:

The third Pythagorean identity that can be arrived by dividing both sides by cos²(θ) in order to get the identity

1 + tan²(θ) = sec²(θ)

The Pythagorean identities are very useful for simplifying the complicated trig statements and equations.

Example 1: Use Pythagorean Identities in order to find the missing trigonometric values.

find the value of the other trig functions

Example 2:

Example 3:

Example 4:

If \sin(30^\circ) = \frac{1}{2},sin(30∘)=21​, what is \cos(30^\circ)?cos(30∘)?

By the Pythagorean identity,

sin2(30∘)+cos2(30∘)=1

Hence,

cos2(30∘) = ±√3/2

Example 5:

Simplify  using the Pythagorean identities



This is good, but we can go even further. If  then we can rearrange this identity by moving the 1 to the other side:  and then multiplying both sides by -1:  . Thus, our expression actually equals:

Being able to manipulate expressions will be helpful when solving more complex equations.

Example 6:

Example 7:

Deriving the Pythagorean identities lead to the understanding of the basic trigonometric proofs. Doing so will help them in solving an equation in both sides of the equation which can be manipulated to find a solution. However while proving for an identity; only one side of the identity can be manipulated. Pythagorean identities can be used for rewriting the trigonometric equations in equivalent form.

The Pythagorean Theorem allows an individual to convert between the sine and cosine values of an angle without being aware of the angle itself. Considering the example, the angle θ in quadrant IV for which the sin(θ) = -24/25. Here the Pythogorean identity can be used along with sin(θ) , in order to solve for cos(θ) .

While solving it the required answer would be cos(θ) = 7/25.

Thus Pythagorean identities can be used to simplify trigonometric expressions in terms of writing other trigonometric functions.