AI Tutor
Imagine tracking a satellite with only a horizontal radar sensor that gives you its cosine value. Can you calculate its exact vertical position without any extra tracking equipment?
You can, because sine and cosine are locked together by one of the most famous equations in geometry.

The Pythagorean Identity

In any right triangle with hypotenuse 11, the horizontal leg equals cos⁡(θ)\cos(\theta) and the vertical leg equals sin⁡(θ)\sin(\theta).
Applying the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) gives the fundamental Pythagorean identity:
sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1
The notation sin⁡2(θ)\sin^2(\theta) is just shorthand for (sin⁡(θ))2(\sin(\theta))^2, meaning you calculate the sine ratio first and then square the result.
In the second century BCE, Greek astronomer Hipparchus first tabulated chord lengths in circles, establishing the right-triangle relationships that led to this universal rule.
So if you know cos⁡(θ)\cos(\theta), you can easily solve for sin⁡(θ)\sin(\theta) by rearranging the formula — but what happens when angles rotate beyond 90∘90^\circ?

Choosing the Right Sign

When you solve sin⁡2(θ)=1−cos⁡2(θ)\sin^2(\theta) = 1 - \cos^2(\theta), taking the square root gives two algebraic solutions: a positive value and a negative value.
sin⁡(θ)=±1−cos⁡2(θ)\sin(\theta) = \pm\sqrt{1 - \cos^2(\theta)}