Find all possible values for a such that the function f (x) = cos (x + a) is odd (i.e.
f (−x) = − f (x) for all x ∈ D f )

To determine the values of \( a \) for which the function \( f(x) = \cos(x + a) \) is odd, we need to ensure that the condition \( f(-x) = -f(x) \) holds for all \( x \).

### Step 1: Express the condition for oddness
The function \( f(x) = \cos(x + a) \) is odd if:
\[
f(-x) = -f(x)
\]
Substitute \( x \) and \( -x \) into the function:
\[
f(-x) = \cos(-x + a)
\]
We know that the cosine function has the property \( \cos(-\theta) = \cos(\theta) \), so:
\[
f(-x) = \cos(-x + a) = \cos(x - a)
\]
Therefore, the condition \( f(-x) = -f(x) \) becomes:
\[
\cos(x - a) = -\cos(x + a)
\]

### Step 2: Use trigonometric identities
Recall that:
\[
-\cos(x + a) = \cos(\pi + x + a)
\]
So, we need:
\[
\cos(x - a) = \cos(\pi + x + a)
\]

### Step 3: Solve for \( a \)
For the equation \( \cos(\theta_1) = \cos(\theta_2) \) to hold, \( \theta_1 \) and \( \theta_2 \) must either be equal modulo \( 2\pi \) or differ by \( 2k\pi \), where \( k \) is an integer. This gives us two cases:

1. **Case 1:** \( x - a = \pi + x + a + 2k\pi \)
   \[
   -a = \pi + a + 2k\pi \implies -2a = \pi + 2k\pi \implies a = -\frac{\pi(1 + 2k)}{2}
   \]
   
2. **Case 2:** \( x - a = -(\pi + x + a) + 2m\pi \)
   \[
   x - a = -\pi - x - a + 2m\pi \implies 2x = -\pi + 2m\pi
   \]
   This equation does not yield a solution for \( a \) independent of \( x \), so it doesn't apply.

### Conclusion:
The function \( f(x) = \cos(x + a) \) is odd if:
\[
a = -\frac{\pi(1 + 2k)}{2}
\]
for any integer \( k \).

This means \( a \) must be of the form \( a = -\frac{\pi}{2}, -\frac{3\pi}{2}, -\frac{5\pi}{2}, \) and so on.

### Would you like further details or have any questions?

### Related Questions:
1. What is the effect of shifting \( a \) on the symmetry of \( \cos(x + a) \)?
2. How does the cosine function's even property affect the oddness of \( f(x) \)?
3. How does the sine function compare when analyzed for oddness?
4. Can the function \( f(x) = \cos(x + a) + \sin(x + b) \) be made odd? What conditions must \( a \) and \( b \) satisfy?
5. What are the implications for symmetry when \( a \) is a multiple of \( \pi/2 \)?

**Tip:** In trigonometric identities, knowing the properties of sine and cosine, such as their even and odd characteristics, is crucial for solving symmetry-related problems.

Find all possible values for a such that the function f(x) = cos(x + a) is odd (i.e. f(-x) = -f(x) for all x ∈ Df).

Explore how to find the values of a for which the function f(x) = cos(x + a) is odd, using trigonometric identities and properties of cosine functions. 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