The Pythagorean Identity

Now that we can define sine and cosine, we will learn how they relate to each other and the unit circle. Recall that the equation for the unit circle is [latex]{x}^{2}+{y}^{2}=1[/latex]. Because [latex]x=\cos t[/latex] and [latex]y=\sin t[/latex], we can substitute for [latex]x[/latex] and [latex]y[/latex] to get [latex]{\cos }^{2}t+{\sin }^{2}t=1[/latex]. This equation, [latex]{\cos }^{2}t+{\sin }^{2}t=1[/latex], is known as the Pythagorean Identity.

We can use the Pythagorean Identity to find the cosine of an angle if we know the sine, or to find the sine of an angle if we know the cosine. However, because the equation yields two solutions, we need additional knowledge of the angle to choose the solution with the correct sign. If we know the quadrant where the angle is, we can easily choose the correct solution.

If [latex]\sin \left(t\right)=\frac{3}{7}[/latex] and [latex]t[/latex] is in the second quadrant, find [latex]\cos \left(t\right)[/latex].

Show Solution

If we drop a vertical line from the point on the unit circle corresponding to [latex]t[/latex], we create a right triangle, from which we can see that the Pythagorean Identity is simply one case of the Pythagorean Theorem. 
 

Substituting the known value for sine into the Pythagorean Identity,

[latex]\begin{gathered}{\cos }^{2}\left(t\right)+{\sin }^{2}\left(t\right)=1 \\ {\cos }^{2}\left(t\right)+\frac{9}{49}=1 \\ {\cos }^{2}\left(t\right)=\frac{40}{49} \\ \text{cos}\left(t\right)=\pm \sqrt{\frac{40}{49}}=\pm \frac{\sqrt{40}}{7}=\pm \frac{2\sqrt{10}}{7} \end{gathered}[/latex]

Because the angle is in the second quadrant, we know the x-value is a negative real number, so the cosine is also negative. So
[latex]\text{cos}\left(t\right)=-\frac{2\sqrt{10}}{7}\\[/latex]