5. Determine the exact value for the following: (5 marks) $$ \sin \frac{11 \pi}{6} \sec \frac{2 \pi}{3}+\tan \frac{5 \pi}{4} $$

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UpStudy AI · Accepted Answer

To determine the exact value of the expression:

$$
\sin \frac{11\pi}{6} \sec \frac{2\pi}{3} + \tan \frac{5\pi}{4}
$$

we will evaluate each trigonometric function separately.

1. **Evaluating $$\sin \frac{11\pi}{6}$$:**
   - The angle $$\frac{11\pi}{6}$$ is in the **4th quadrant** where sine is negative.
   - The reference angle is $$2\pi - \frac{11\pi}{6} = \frac{\pi}{6}$$.
   - $$\sin \frac{\pi}{6} = \frac{1}{2}$$, so:
     $$
     \sin \frac{11\pi}{6} = -\frac{1}{2}
     $$

2. **Evaluating $$\sec \frac{2\pi}{3}$$:**
   - Recall that $$\sec \theta = \frac{1}{\cos \theta}$$.
   - The angle $$\frac{2\pi}{3}$$ is in the **2nd quadrant** where cosine is negative.
   - The reference angle is $$\pi - \frac{2\pi}{3} = \frac{\pi}{3}$$.
   - $$\cos \frac{\pi}{3} = \frac{1}{2}$$, so:
     $$
     \cos \frac{2\pi}{3} = -\frac{1}{2}
     $$
     $$
     \sec \frac{2\pi}{3} = \frac{1}{\cos \frac{2\pi}{3}} = \frac{1}{-\frac{1}{2}} = -2
     $$

3. **Evaluating $$\tan \frac{5\pi}{4}$$:**
   - The angle $$\frac{5\pi}{4}$$ is in the **3rd quadrant** where tangent is positive.
   - The reference angle is $$\frac{5\pi}{4} - \pi = \frac{\pi}{4}$$.
   - $$\tan \frac{\pi}{4} = 1$$, so:
     $$
     \tan \frac{5\pi}{4} = 1
     $$

4. **Combining the results:**
   $$
   \sin \frac{11\pi}{6} \cdot \sec \frac{2\pi}{3} + \tan \frac{5\pi}{4} = \left(-\frac{1}{2}\right) \cdot (-2) + 1 = 1 + 1 = 2
   $$

**Final Answer:**
$$
2
$$
The exact value of the expression is 2.

Determine the exact value for the following: (5 marks)

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