Find the value of cos2
(x) + cos2

x +
π
3

+ cos2

x −
π
3


We are tasked with finding the value of:

\[
\cos^2(x) + \cos^2\left(x + \frac{\pi}{3}\right) + \cos^2\left(x - \frac{\pi}{3}\right)
\]

### Step 1: Use trigonometric identities
Recall that:

\[
\cos(a + b) = \cos a \cos b - \sin a \sin b
\]

and

\[
\cos(a - b) = \cos a \cos b + \sin a \sin b
\]

Using these identities, we can express \( \cos\left(x + \frac{\pi}{3}\right) \) and \( \cos\left(x - \frac{\pi}{3}\right) \).

### Step 2: Evaluate \( \cos\left(x + \frac{\pi}{3}\right) \) and \( \cos\left(x - \frac{\pi}{3}\right) \)
Since \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \) and \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \), we get the following:

\[
\cos\left(x + \frac{\pi}{3}\right) = \cos x \cdot \frac{1}{2} - \sin x \cdot \frac{\sqrt{3}}{2}
\]
\[
\cos\left(x - \frac{\pi}{3}\right) = \cos x \cdot \frac{1}{2} + \sin x \cdot \frac{\sqrt{3}}{2}
\]

### Step 3: Square each term
Now, let's square these results:

\[
\cos^2\left(x + \frac{\pi}{3}\right) = \left(\frac{1}{2} \cos x - \frac{\sqrt{3}}{2} \sin x\right)^2 = \frac{1}{4} \cos^2 x - \frac{\sqrt{3}}{2} \cos x \sin x + \frac{3}{4} \sin^2 x
\]
\[
\cos^2\left(x - \frac{\pi}{3}\right) = \left(\frac{1}{2} \cos x + \frac{\sqrt{3}}{2} \sin x\right)^2 = \frac{1}{4} \cos^2 x + \frac{\sqrt{3}}{2} \cos x \sin x + \frac{3}{4} \sin^2 x
\]

### Step 4: Add all the expressions
Now we can add up all the terms:

\[
\cos^2(x) + \cos^2\left(x + \frac{\pi}{3}\right) + \cos^2\left(x - \frac{\pi}{3}\right)
\]

Substituting the squared terms:

\[
= \cos^2 x + \left( \frac{1}{4} \cos^2 x - \frac{\sqrt{3}}{2} \cos x \sin x + \frac{3}{4} \sin^2 x \right) + \left( \frac{1}{4} \cos^2 x + \frac{\sqrt{3}}{2} \cos x \sin x + \frac{3}{4} \sin^2 x \right)
\]

Simplifying:

- The mixed terms \( - \frac{\sqrt{3}}{2} \cos x \sin x \) and \( + \frac{\sqrt{3}}{2} \cos x \sin x \) cancel each other out.
- The remaining terms give:

\[
= \cos^2 x + \frac{1}{4} \cos^2 x + \frac{1}{4} \cos^2 x + \frac{3}{4} \sin^2 x + \frac{3}{4} \sin^2 x
\]
\[
= \cos^2 x + \frac{1}{2} \cos^2 x + \frac{3}{2} \sin^2 x
\]
\[
= \frac{3}{2} \cos^2 x + \frac{3}{2} \sin^2 x
\]

Since \( \cos^2 x + \sin^2 x = 1 \), we have:

\[
= \frac{3}{2} \cdot 1 = \frac{3}{2}
\]

Thus, the value of the expression is:

\[
\boxed{\frac{3}{2}}
\]

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. What is the double-angle identity for cosine, and how could it help simplify expressions?
2. How would the result change if the angle shifts were \( \frac{\pi}{4} \) instead of \( \frac{\pi}{3} \)?
3. How do you derive trigonometric identities using Euler's formula?
4. Can you apply the same method to solve for similar trigonometric sums involving sine?
5. What are the key differences between the sine and cosine angle sum identities?

**Tip:** Always remember that \( \cos^2 x + \sin^2 x = 1 \) is a fundamental identity and can simplify many expressions!

Find the value of cos²(x) + cos²(x + π/3) + cos²(x - π/3)

Explore how to calculate cos²(x) + cos²(x + π/3) + cos²(x - π/3) using trigonometric identities. Learn step-by-step how to apply the angle sum and difference identities and the Pythagorean identity to simplify this expression. This problem is ideal for 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