Question

3 matching 20 points
find the exact value of each trigonometric function given the following angle measure (θ):
( \tan \theta = )
( sin \theta = )
( cos \theta = )
4 multiple answer 20 points

Explanation:

Step1: Determine the reference angle

The given angle \(\theta=\pi-\frac{\pi}{3}=\frac{2\pi}{3}\). The reference angle \(\theta'=\pi - \frac{2\pi}{3}=\frac{\pi}{3}\)

Step2: Find \(\sin\theta\)

Using the formula \(\sin\theta=\sin(\pi - \alpha)=\sin\alpha\) (where \(\alpha = \frac{\pi}{3}\)). Since \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\), and \(\theta=\frac{2\pi}{3}\) is in the second - quadrant where \(\sin\theta>0\), so \(\sin\theta=\frac{\sqrt{3}}{2}\)

Step3: Find \(\cos\theta\)

Using the formula \(\cos\theta=-\cos\alpha\) (because \(\theta\) is in the second - quadrant and \(\cos\theta<0\) when \(\theta\in(\frac{\pi}{2},\pi)\), and \(\alpha=\frac{\pi}{3}\)). Since \(\cos\frac{\pi}{3}=\frac{1}{2}\), so \(\cos\theta =-\frac{1}{2}\)

Step4: Find \(\tan\theta\)

Using the formula \(\tan\theta=-\tan\alpha\) (because \(\theta\) is in the second - quadrant and \(\tan\theta<0\) when \(\theta\in(\frac{\pi}{2},\pi)\), and \(\alpha = \frac{\pi}{3}\)). Since \(\tan\frac{\pi}{3}=\sqrt{3}\), so \(\tan\theta=-\sqrt{3}\)

Answer:

\(\tan\theta=-\sqrt{3}\), \(\sin\theta=\frac{\sqrt{3}}{2}\), \(\cos\theta =-\frac{1}{2}\)