Let's solve each of the trigonometric functions:

a) $\csc\left(\frac{3\pi}{4}\right)$

• The cosecant function is the reciprocal of the sine:
  $\csc(\theta) = \frac{1}{\sin(\theta)}$
• First, determine $\sin\left(\frac{3\pi}{4}\right)$. The angle $\frac{3\pi}{4}$ lies in the second quadrant, where sine is positive.
• $\sin\left(\frac{3\pi}{4}\right) = \sin\left(\pi - \frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

Thus: $\csc\left(\frac{3\pi}{4}\right) = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$

b) $\tan\left(-\frac{5\pi}{6}\right)$

• The tangent function is periodic with a period of $\pi$, and for negative angles, we reflect across the x-axis.
• $\tan\left(-\frac{5\pi}{6}\right) = -\tan\left(\frac{5\pi}{6}\right)$
• The angle $\frac{5\pi}{6}$ is in the second quadrant, where tangent is negative.
  $\tan\left(\frac{5\pi}{6}\right) = \tan\left(\pi - \frac{\pi}{6}\right) = -\tan\left(\frac{\pi}{6}\right) = -\frac{1}{\sqrt{3}}$
• Therefore: $\tan\left(-\frac{5\pi}{6}\right) = \frac{1}{\sqrt{3}}$

c) $\sec\left(-\frac{7\pi}{3}\right)$

• The secant function is the reciprocal of the cosine: $\sec(\theta) = \frac{1}{\cos(\theta)}$
• To simplify $\sec\left(-\frac{7\pi}{3}\right)$, use the periodicity of cosine (period $2\pi$).
• $-\frac{7\pi}{3}$ can be rewritten as:
  $-\frac{7\pi}{3} + 2\pi = -\frac{7\pi}{3} + \frac{6\pi}{3} = -\frac{\pi}{3}$
• $\sec\left(-\frac{\pi}{3}\right) = \sec\left(\frac{\pi}{3}\right)$ (cosine is an even function).
• $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$

Thus: $\sec\left(-\frac{\pi}{3}\right) = \frac{1}{\frac{1}{2}} = 2$

Final answers:

a) $\csc\left(\frac{3\pi}{4}\right) = \sqrt{2}$
b) $\tan\left(-\frac{5\pi}{6}\right) = \frac{1}{\sqrt{3}}$
c) $\sec\left(-\frac{7\pi}{3}\right) = 2$

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Would you like further details or have any questions on these?
Here are some related questions:

1. What is the value of $\sin\left(\frac{5\pi}{6}\right)$?
2. How do you find the period of trigonometric functions like secant or cosecant?
3. Why is tangent negative in the second quadrant?
4. What is the relationship between cotangent and tangent?
5. Can you explain how trigonometric identities apply to negative angles?

Tip: When simplifying trigonometric functions with large angles, remember to reduce them using the periodicity of the function.