Question

find the exact values of the six trigonometric functions of the angle. - 330°
sin(-330°)=□
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Find the coterminal positive angle

Add $360^{\circ}$ to $- 330^{\circ}$. So, $-330^{\circ}+360^{\circ}=30^{\circ}$.

Step2: Recall sine - angle property

Since $\sin(-\theta)=-\sin\theta$ and for $\theta = 330^{\circ}$, we know that $\sin(-330^{\circ})=\sin(30^{\circ})$.
We know from the unit - circle or special right - triangles that $\sin(30^{\circ})=\frac{1}{2}$.

Step3: Find cosine value

$\cos(-330^{\circ})=\cos(30^{\circ})=\frac{\sqrt{3}}{2}$ (because $\cos(-\theta)=\cos\theta$).

Step4: Find tangent value

$\tan(-330^{\circ})=\tan(30^{\circ})=\frac{\sin(30^{\circ})}{\cos(30^{\circ})}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$.

Step5: Find cosecant value

$\csc(-330^{\circ})=\frac{1}{\sin(-330^{\circ})}=\frac{1}{\frac{1}{2}} = 2$.

Step6: Find secant value

$\sec(-330^{\circ})=\frac{1}{\cos(-330^{\circ})}=\frac{1}{\frac{\sqrt{3}}{2}}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}$.

Step7: Find cotangent value

$\cot(-330^{\circ})=\frac{1}{\tan(-330^{\circ})}=\frac{1}{\frac{\sqrt{3}}{3}}=\sqrt{3}$.

Answer:

$\sin(-330^{\circ})=\frac{1}{2}$, $\cos(-330^{\circ})=\frac{\sqrt{3}}{2}$, $\tan(-330^{\circ})=\frac{\sqrt{3}}{3}$, $\csc(-330^{\circ}) = 2$, $\sec(-330^{\circ})=\frac{2\sqrt{3}}{3}$, $\cot(-330^{\circ})=\sqrt{3}$