Question 4.2.6: Using a Pythagorean Identity Given that sin t = 3/5 and 0 ≤ ...
Using a Pythagorean Identity
Given that \sin t=\frac{3}{5} and 0 \leq t<\frac{\pi}{2}, find the value of cos t using a trigonometric identity.
We can find the value of cos t by using the Pythagorean identity
\sin ^2 t+\cos ^2 t=1.
\left(\frac{3}{5}\right)^2+\cos ^2 t=1 We are given that \sin t=\frac{3}{5}.
\frac{9}{25}+\cos ^2 t=1 Square \frac{3}{5}:\left(\frac{3}{5}\right)^2=\frac{3^2}{5^2}=\frac{9}{25}.
\cos ^2 t=1-\frac{9}{25} Subtract \frac{9}{25} from both sides.
\cos ^2 t=\frac{16}{25} Simplify: 1-\frac{9}{25}=\frac{25}{25}-\frac{9}{25}=\frac{16}{25}.
\cos t=\sqrt{\frac{16}{25}}=\frac{4}{5} Because O \leq t<\frac{\pi}{2}, \cos t, the x-coordinate of a point on the unit circle, is positive.
Thus, \cos t=\frac{4}{5}.