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Pythagorean Sine and Cosine Identities on a Unit Circle

If you have ever wondered why the Pythagorean identity, sin2θ + cos2θ = 1, is so important, and where it came from, then read on. This identity is important because it sets an expression involving trig functions equal to 1, and this simplification is very helpful for solving equations. As such, this is probably one of the most frequently used trig identities.

Finding the Pythagorean identity on a unit circle.
Finding the Pythagorean identity on a unit circle.

As you can see in the preceding figure, this identity comes from putting a right triangle inside the unit circle and substituting values and equations to come up with a whole new equation.

image1.png

point and r is the radius of the circle. The value of x is also the length of the adjacent side of the triangle, and y is the length of the opposite side. In a unit circle, the radius is equal to 1. When you substitute that value in the equation, you find that

image2.png

Hold that thought.

The Pythagorean theorem says that when you square the value of each of a triangle’s two legs and add the results together, you get the square of the hypotenuse. In mathematical notation, it looks like this: a2 + b2 = c2. In the case of the right triangle on the unit circle, because the radius (which is also the hypotenuse) is 1, you can say that x2 + y2 = 12. Now replace the x with cos and the y with sin, switch the two terms around, and you get sin2 + cos2 = 1.

If all the finagling just seems like a lot of hocus pocus to you, check out this identity in action. Suppose the angle in question is 30 degrees. Using the values for the functions of a 30-degree angle,

image3.png

and putting them into the identity, you get

image4.png

Voilà!

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