Cos²30°+ cos²60°+cos²90°+2cos30°cos60°cos90°

We will simplify the given expression:

\[
\cos^2 30^\circ + \cos^2 60^\circ + \cos^2 90^\circ + 2 \cos 30^\circ \cos 60^\circ \cos 90^\circ
\]

### Step-by-step solution:

1. **Evaluate each cosine term:**
   - \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
   - \(\cos 60^\circ = \frac{1}{2}\)
   - \(\cos 90^\circ = 0\)

2. **Square the cosine terms:**
   \[
   \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}
   \]
   \[
   \cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
   \]
   \[
   \cos^2 90^\circ = 0^2 = 0
   \]

3. **Substitute the squared values into the expression:**
   \[
   \frac{3}{4} + \frac{1}{4} + 0 + 2 \cdot \cos 30^\circ \cdot \cos 60^\circ \cdot \cos 90^\circ
   \]

4. **Simplify the remaining cosine product:**
   Since \(\cos 90^\circ = 0\), the product \(2 \cdot \cos 30^\circ \cdot \cos 60^\circ \cdot \cos 90^\circ = 0\).

5. **Final simplification:**
   \[
   \frac{3}{4} + \frac{1}{4} + 0 = 1
   \]

Thus, the value of the given expression is:
\[
\boxed{1}
\]

Would you like more details or have any questions about this process?

Here are some related questions to consider:
1. How do you calculate other trigonometric values like \(\sin 30^\circ\) or \(\tan 60^\circ\)?
2. What are the cosine values for angles other than standard angles like 45°?
3. How can the cosine addition and subtraction formulas be applied?
4. What are the applications of cosine laws in geometry and physics?
5. How does the unit circle help in understanding trigonometric values?

**Tip:** Memorizing values of trigonometric functions at key angles like 0°, 30°, 45°, 60°, and 90° helps in quickly solving many trigonometric problems.

Cos²30° + cos²60° + cos²90° + 2cos30°cos60°cos90°

Explore the step-by-step solution to the trigonometric expression Cos²30° + cos²60° + cos²90° + 2cos30°cos60°cos90°. Learn about cosine values at standard angles and how to simplify using key trigonometric identities. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation