We will solve the expression \( \cos 144^\circ \cos 24^\circ + \sin 144^\circ \cos 24^\circ \).

### Step-by-step solution:

1. **Group the terms:**

   We can factor out \( \cos 24^\circ \) from both terms:
   \[
   \cos 144^\circ \cos 24^\circ + \sin 144^\circ \cos 24^\circ = (\cos 144^\circ + \sin 144^\circ) \cos 24^\circ
   \]

2. **Simplify using trigonometric values:**
   We now compute the trigonometric values for \( \cos 144^\circ \) and \( \sin 144^\circ \):
   - \( \cos 144^\circ = -\cos 36^\circ \)
   - \( \sin 144^\circ = \sin 36^\circ \)

   Therefore, the expression becomes:
   \[
   (-\cos 36^\circ + \sin 36^\circ) \cos 24^\circ
   \]

3. **Substitute known values for cosine and sine of 36°:**
   The exact values are:
   \[
   \cos 36^\circ = \frac{\sqrt{5}+1}{4}, \quad \sin 36^\circ = \frac{\sqrt{10 - 2\sqrt{5}}}{4}
   \]

4. **Final Expression:**
   Now, the simplified result is:
   \[
   (-\cos 36^\circ + \sin 36^\circ) \cos 24^\circ
   \]
   which can be evaluated numerically if needed.

Would you like me to evaluate this numerically?

---

Here are some related questions:

1. What is the general formula for \( \cos(A) \cos(B) + \sin(A) \cos(B) \)?
2. Can you prove the exact trigonometric values of special angles like \( \cos 36^\circ \)?
3. How do sum and difference identities apply to this trigonometric expression?
4. What are the trigonometric values for angles like \( 24^\circ \) and \( 144^\circ \)?
5. How would you graph \( \cos 144^\circ \cos 24^\circ + \sin 144^\circ \cos 24^\circ \)?

**Tip:** It's helpful to memorize the values of trigonometric functions for special angles, as they often simplify complex expressions.

Learn how to simplify the trigonometric expression cos(144°)cos(24°) + sin(144°)cos(24°) step-by-step, using angle addition formulas and exact trigonometric values for special angles. This solution is ideal for high school students in Grades 10-12 studying advanced trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation