3.2 Simplify $$ 3.21 \frac{\operatorname{Sin} 550}{\operatorname{Cos}(-170)} $$

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UpStudy AI · Accepted Answer

To simplify the expression:

$$
3.21 \times \frac{\sin 550^\circ}{\cos (-170^\circ)}
$$

follow these steps:

1. **Simplify the Angles:**

- **Sine Function:**
     $$
     \sin 550^\circ = \sin (360^\circ + 190^\circ) = \sin 190^\circ
     $$
     Using the sine addition formula:
     $$
     \sin 190^\circ = \sin (180^\circ + 10^\circ) = -\sin 10^\circ
     $$
   
   - **Cosine Function:**
     $$
     \cos (-170^\circ) = \cos 170^\circ
     $$
     Since cosine is an even function:
     $$
     \cos 170^\circ = \cos (180^\circ - 10^\circ) = -\cos 10^\circ
     $$

2. **Substitute the Simplified Values:**
   $$
   \frac{\sin 550^\circ}{\cos (-170^\circ)} = \frac{-\sin 10^\circ}{-\cos 10^\circ} = \frac{\sin 10^\circ}{\cos 10^\circ} = \tan 10^\circ
   $$

3. **Multiply by 3.21:**
   $$
   3.21 \times \tan 10^\circ
   $$
   
4. **Calculate the Numerical Value:**
   
   - First, find $$\tan 10^\circ$$:
     $$
     \tan 10^\circ \approx 0.176327
     $$
   
   - Then, multiply by 3.21:
     $$
     3.21 \times 0.176327 \approx 0.566
     $$

**Final Simplified Value:**
$$
\boxed{0.566}
$$
The simplified value of the expression is approximately 0.566.

3.2 Simplify \( 3.21 \frac{\operatorname{Sin} 550}{\operatorname{Cos}(-170)} \)

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