Question

video assignment 7.1 angles
score: 9/13 answered: 9/13
question 10
if $\theta=\frac{2\pi}{3}$, then
$\sin(\theta)$ equals
$\cos(\theta)$ equals
$\tan(\theta)$ equals
$\sec(\theta)$ equals
question help: worked example 1

Explanation:

Step1: Find \(\sin(\theta)\)

We know that \(\theta=\frac{2\pi}{3}\). Using the unit - circle or the formula \(\sin(A + B)=\sin A\cos B+\cos A\sin B\) (where \(A = \pi-\frac{\pi}{3}\)), and \(\sin(\pi - x)=\sin x\). So \(\sin(\frac{2\pi}{3})=\sin(\pi-\frac{\pi}{3})=\sin(\frac{\pi}{3})\). Since \(\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}\), then \(\sin(\frac{2\pi}{3})=\frac{\sqrt{3}}{2}\).

Step2: Find \(\cos(\theta)\)

Using the formula \(\cos(\pi - x)=-\cos x\). For \(\theta=\frac{2\pi}{3}=\pi-\frac{\pi}{3}\), \(\cos(\frac{2\pi}{3})=\cos(\pi - \frac{\pi}{3})=-\cos(\frac{\pi}{3})\). Since \(\cos(\frac{\pi}{3})=\frac{1}{2}\), then \(\cos(\frac{2\pi}{3})=-\frac{1}{2}\).

Step3: Find \(\tan(\theta)\)

Using the formula \(\tan\theta=\frac{\sin\theta}{\cos\theta}\). Substitute \(\sin\theta=\frac{\sqrt{3}}{2}\) and \(\cos\theta =-\frac{1}{2}\). Then \(\tan(\frac{2\pi}{3})=\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}=-\sqrt{3}\).

Step4: Find \(\sec(\theta)\)

Using the formula \(\sec\theta=\frac{1}{\cos\theta}\). Substitute \(\cos\theta =-\frac{1}{2}\). Then \(\sec(\frac{2\pi}{3})=\frac{1}{-\frac{1}{2}}=- 2\).

Answer:

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