Let's go through each trigonometric expression one by one to find their exact values:

a. $\sec 60^\circ$

$\sec \theta = \frac{1}{\cos \theta}$ Since $\cos 60^\circ = \frac{1}{2}$, $\sec 60^\circ = \frac{1}{\frac{1}{2}} = 2$

b. $\csc 45^\circ$

$\csc \theta = \frac{1}{\sin \theta}$ Since $\sin 45^\circ = \frac{\sqrt{2}}{2}$, $\csc 45^\circ = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$

c. $\cot 120^\circ$

$\cot \theta = \frac{1}{\tan \theta}$ First, find $\tan 120^\circ$. Since $\tan 120^\circ = -\tan 60^\circ = -\sqrt{3}$, $\cot 120^\circ = \frac{1}{-\sqrt{3}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

d. $\sec 300^\circ$

$\sec 300^\circ = \frac{1}{\cos 300^\circ}$ Since $\cos 300^\circ = \cos (360^\circ - 60^\circ) = \cos 60^\circ = \frac{1}{2}$, $\sec 300^\circ = \frac{1}{\frac{1}{2}} = 2$

e. $\csc 135^\circ$

$\csc 135^\circ = \frac{1}{\sin 135^\circ}$ Since $\sin 135^\circ = \sin (180^\circ - 45^\circ) = \sin 45^\circ = \frac{\sqrt{2}}{2}$, $\csc 135^\circ = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$

f. $\cot 330^\circ$

$\cot 330^\circ = \frac{1}{\tan 330^\circ}$ Since $\tan 330^\circ = \tan (360^\circ - 30^\circ) = -\tan 30^\circ = -\frac{1}{\sqrt{3}}$, $\cot 330^\circ = \frac{1}{-\frac{1}{\sqrt{3}}} = -\sqrt{3}$

g. $\sec 150^\circ$

$\sec 150^\circ = \frac{1}{\cos 150^\circ}$ Since $\cos 150^\circ = \cos (180^\circ - 30^\circ) = -\cos 30^\circ = -\frac{\sqrt{3}}{2}$, $\sec 150^\circ = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}$

h. $\cot(-30^\circ)$

$\cot(-30^\circ) = \frac{1}{\tan(-30^\circ)}$ Since $\tan(-30^\circ) = -\tan 30^\circ = -\frac{1}{\sqrt{3}}$, $\cot(-30^\circ) = \frac{1}{-\frac{1}{\sqrt{3}}} = -\sqrt{3}$

Final Answers:

• a. $\sec 60^\circ = 2$
• b. $\csc 45^\circ = \sqrt{2}$
• c. $\cot 120^\circ = -\frac{\sqrt{3}}{3}$
• d. $\sec 300^\circ = 2$
• e. $\csc 135^\circ = \sqrt{2}$
• f. $\cot 330^\circ = -\sqrt{3}$
• g. $\sec 150^\circ = -\frac{2\sqrt{3}}{3}$
• h. $\cot(-30^\circ) = -\sqrt{3}$

Would you like further details or have any questions?

Related Questions:

1. What is the exact value of $\sin 210^\circ$?
2. How would you compute $\tan 240^\circ$?
3. Can you explain why $\cot 90^\circ$ is undefined?
4. What is the value of $\sec 270^\circ$ and why?
5. How do you determine which quadrant an angle falls into for trigonometric evaluations?

Tip:

Remember that trigonometric functions repeat every $360^\circ$ or $2\pi$ radians, which can simplify the calculation of angles larger than $360^\circ$.