https://file.aiquickdraw.com/mathbot/file/1727413672888ih0pfudj.jpg

The mathematical expression in the image appears to be a trigonometric equation written in Farsi, with the denominator and numerator containing various trigonometric functions and angles.

Here's a breakdown of the equation in the image:

\[
\frac{\sin(45^\circ) + \cos(180^\circ) - 3\cot(30^\circ)}{4\cos(0^\circ) + \sqrt{3}\sin(45^\circ)\cos(45^\circ)}
\]

### Let's compute it step by step:

#### 1. Numerator:
\[
\sin(45^\circ) + \cos(180^\circ) - 3\cot(30^\circ)
\]

- \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)
- \(\cos(180^\circ) = -1\)
- \(\cot(30^\circ) = \frac{1}{\tan(30^\circ)} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3}\)

So, the numerator becomes:
\[
\frac{\sqrt{2}}{2} + (-1) - 3\sqrt{3}
\]

#### 2. Denominator:
\[
4\cos(0^\circ) + \sqrt{3}\sin(45^\circ)\cos(45^\circ)
\]

- \(\cos(0^\circ) = 1\)
- \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)
- \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\)

So, the denominator becomes:
\[
4(1) + \sqrt{3}\left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)
= 4 + \sqrt{3}\left(\frac{2}{4}\right) = 4 + \frac{\sqrt{3}}{2}
\]

### Final Expression:
\[
\frac{\frac{\sqrt{2}}{2} - 1 - 3\sqrt{3}}{4 + \frac{\sqrt{3}}{2}}
\]

Would you like me to further simplify this result or solve it numerically? Let me know if you need more details!

---

### Related Questions:
1. How do we simplify trigonometric expressions like \(\sin(45^\circ)\) and \(\cos(180^\circ)\)?
2. Why is \(\cot(30^\circ) = \sqrt{3}\), and how does that relate to \(\tan(30^\circ)\)?
3. What are some other examples of simplifying trigonometric identities?
4. How do special angles in trigonometry (like 45° and 30°) make calculations easier?
5. What are the key properties of trigonometric functions that help in solving these problems?

### Tip:
When working with trigonometric expressions, always refer to standard angle values (like 0°, 30°, 45°, 60°, 90°) as they have easily recognizable sine, cosine, and tangent values that can help in quick simplifications.

What is the result of the following trigonometric expression: (Sin 45° + Cos 180° - 3Cot 30°) / (4Cos 0° + √3Sin 45° Cos 45°)?

Solve the trigonometric expression (Sin 45° + Cos 180° - 3Cot 30°) / (4Cos 0° + √3Sin 45° Cos 45°) step by step using values for special angles like 45°, 180°, 30°, and 0°. Learn how to simplify trigonometric expressions involving common angle values.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation