https://file.aiquickdraw.com/mathbot/file/1729044094473yvmwjn4t.jpg

The question in the image is a trigonometric expression. The problem states:

- Given that \(\tan \alpha^\circ = -\) (implying \(\tan \alpha^\circ\) is negative) and \(\alpha^\circ\) is in the second quadrant.
- We need to calculate the following expression:

\[
\frac{\sin(90^\circ - \alpha^\circ) - \cos(180^\circ - \alpha^\circ)}{\tan(270^\circ + \alpha^\circ) + \cot(360^\circ - \alpha^\circ)}
\]

### Step-by-step solution:

1. **Identify trigonometric identities in terms of \(\alpha^\circ\):**
   - \(\sin(90^\circ - \alpha^\circ) = \cos(\alpha^\circ)\) (co-function identity)
   - \(\cos(180^\circ - \alpha^\circ) = -\cos(\alpha^\circ)\) (because \(\cos\) is negative in the second quadrant)
   - \(\tan(270^\circ + \alpha^\circ) = \cot(\alpha^\circ)\) (since \(\tan(270^\circ + x) = \cot(x)\))
   - \(\cot(360^\circ - \alpha^\circ) = \cot(\alpha^\circ)\) (as \(\cot\) repeats every 180 degrees)

2. **Substitute the identities:**
   The expression simplifies to:

   \[
   \frac{\cos(\alpha^\circ) - (-\cos(\alpha^\circ))}{\cot(\alpha^\circ) + \cot(\alpha^\circ)}
   \]
   
   This simplifies to:

   \[
   \frac{\cos(\alpha^\circ) + \cos(\alpha^\circ)}{2 \cdot \cot(\alpha^\circ)}
   \]
   
   Which further simplifies to:

   \[
   \frac{2\cos(\alpha^\circ)}{2 \cdot \cot(\alpha^\circ)} = \frac{\cos(\alpha^\circ)}{\cot(\alpha^\circ)}
   \]

3. **Simplify the final result:**
   Since \(\frac{\cos(\alpha^\circ)}{\cot(\alpha^\circ)} = \sin(\alpha^\circ)\), the final answer is:

   \[
   \sin(\alpha^\circ)
   \]

Would you like further clarification on this solution or details on any specific steps?

---

Here are 5 follow-up questions that could help expand your understanding:

1. Why is \(\tan \alpha^\circ\) negative in the second quadrant?
2. Can you explain how the co-function identities are derived?
3. How does the periodicity of trigonometric functions help simplify expressions?
4. What is the relationship between \(\tan\) and \(\cot\)?
5. How would this problem change if \(\alpha^\circ\) were in a different quadrant?

**Tip:** Always verify the quadrant of an angle when solving trigonometric problems to ensure correct sign usage for sine, cosine, and tangent.

Diketahui tan α° = - dan α° sudut di kuadran II. Hitunglah: (sin(90° − α°) − cos(180° − α°)) / (tan(270° + α°) + cot(360° − α°))

Learn how to solve a trigonometric expression involving tan α° in quadrant II using co-function identities and quadrant-based sign rules. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation