Question

Find the exact value of the trigonometric functions at the indicated angle. \(\sin \theta, \cos \theta\), and \(\tan \theta\) for \(\theta=\pi / 3\)

Short answer

\(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\), \(\cos(\frac{\pi}{3}) = \frac{1}{2}\), and \(\tan(\frac{\pi}{3}) = \sqrt{3}\)

Step-by-step solution

Identify the angle on the unit circle

We are given the angle \(\theta = \pi / 3\). Since one full rotation around the unit circle equals \(2 \pi\) radians, we can determine that this angle corresponds to a point on the circle that is located in the first quadrant.

Determine the coordinates of the point on the unit circle

At the angle \(\theta = \pi / 3\), the coordinates on the unit circle can be expressed as \((x, y)\), where \(x\) is the cosine of the angle and \(y\) is the sine of the angle. Recall that for some common angles, we can directly determine the coordinates (sine and cosine values). In this case, since \(\theta = \pi / 3\) (also referred as a 60 degree angle), we have \(x = \frac{1}{2}\) and \(y = \frac{\sqrt{3}}{2}\).

Evaluate the sine, cosine, and tangent functions

Now that we found the coordinates on the unit circle, we can find the sine, cosine, and tangent values for this angle. We have: - \(\sin(\theta) = y = \frac{\sqrt{3}}{2}\) - \(\cos(\theta) = x = \frac{1}{2}\) - \(\tan(\theta) = \frac{y}{x} =\frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} =\sqrt{3}\)

Present the final values of the trigonometric functions

We found the exact values of the trigonometric functions for the angle \(\theta = \frac{\pi}{3}\): - \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\) - \(\cos(\frac{\pi}{3}) = \frac{1}{2}\) - \(\tan(\frac{\pi}{3}) = \sqrt{3}\)