use the unit circle to determine the exact value of each trig function. 1. sec 35° 2. cot 30° 3. sec 7π/6 4…

use the unit circle to determine the exact value of each trig function. 1. sec 35° 2. cot 30° 3. sec 7π/6 4. cos 80° 5. cos 240° 6. csc 4π/3 7. cos 45° 8. sin 3π/2 9. tan 2π/3 10. sin π/6 11. csc 180° 12. sec π/4 13. cos 300° 14. sec 3π/4 15. tan 35° 16. cos 2π/3 17. cot π/2 18. tan 30°

Answer

Explanation:

Step1: Recall trigonometric identities

$\sec\theta=\frac{1}{\cos\theta}$, $\cot\theta = \frac{\cos\theta}{\sin\theta}$, $\csc\theta=\frac{1}{\sin\theta}$, $\tan\theta=\frac{\sin\theta}{\cos\theta}$

Step2: Use unit - circle values

For $\theta = 30^{\circ}$, $\sin30^{\circ}=\frac{1}{2}$, $\cos30^{\circ}=\frac{\sqrt{3}}{2}$; for $\theta = 45^{\circ}$, $\sin45^{\circ}=\cos45^{\circ}=\frac{\sqrt{2}}{2}$; for $\theta=60^{\circ}$, $\sin60^{\circ}=\frac{\sqrt{3}}{2}$, $\cos60^{\circ}=\frac{1}{2}$; for angles in radians, convert to degrees if needed ($\pi$ radians = 180°).

$\sec30^{\circ}=\frac{1}{\cos30^{\circ}}=\frac{2\sqrt{3}}{3}$
$\cot30^{\circ}=\frac{\cos30^{\circ}}{\sin30^{\circ}}=\sqrt{3}$
$\sec\frac{7\pi}{6}=\frac{1}{\cos\frac{7\pi}{6}}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2\sqrt{3}}{3}$
$\cos80^{\circ}\approx0.174$ (using calculator, exact value from unit - circle is complex without further approximation)
$\cos240^{\circ}=-\frac{1}{2}$
$\csc\frac{4\pi}{3}=\frac{1}{\sin\frac{4\pi}{3}}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2\sqrt{3}}{3}$
$\cos45^{\circ}=\frac{\sqrt{2}}{2}$
$\sin\frac{3\pi}{2}=- 1$
$\tan\frac{2\pi}{3}=\frac{\sin\frac{2\pi}{3}}{\cos\frac{2\pi}{3}}=\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}=-\sqrt{3}$
$\sin\frac{\pi}{6}=\frac{1}{2}$
$\csc180^{\circ}=\frac{1}{\sin180^{\circ}}$, and $\sin180^{\circ} = 0$, so $\csc180^{\circ}$ is undefined.
$\sec\frac{\pi}{4}=\frac{1}{\cos\frac{\pi}{4}}=\sqrt{2}$
$\cos300^{\circ}=\frac{1}{2}$
$\sec\frac{3\pi}{4}=\frac{1}{\cos\frac{3\pi}{4}}=\frac{1}{-\frac{\sqrt{2}}{2}}=-\sqrt{2}$
$\tan35^{\circ}\approx0.700$ (using calculator, exact value from unit - circle is complex without further approximation)
$\cos\frac{2\pi}{3}=-\frac{1}{2}$
$\cot\frac{\pi}{2}=\frac{\cos\frac{\pi}{2}}{\sin\frac{\pi}{2}} = 0$
$\tan30^{\circ}=\frac{\sqrt{3}}{3}$

Answer:

$\frac{2\sqrt{3}}{3}$
$\sqrt{3}$
$-\frac{2\sqrt{3}}{3}$
$\approx0.174$
$-\frac{1}{2}$
$-\frac{2\sqrt{3}}{3}$
$\frac{\sqrt{2}}{2}$
$-1$
$-\sqrt{3}$
$\frac{1}{2}$
Undefined
$\sqrt{2}$
$\frac{1}{2}$
$-\sqrt{2}$
$\approx0.700$
$-\frac{1}{2}$
$0$
$\frac{\sqrt{3}}{3}$