If `cos40^(@)=xandcostheta=1-2x^(2)`, then the possible values of `theta` lying between `0^(@)` and `360^(@)` is

Options

A. `100^(@)and260^(@)`
B. `80^(@)and280^(@)`
C. `280^(@)and110^(@)`
D. `110^(@)and260^(@)`

Answer

To solve the problem, we start with the given equations:

1. \( \cos 40^\circ = x \)
2. \( \cos \theta = 1 - 2x^2 \)

### Step 1: Substitute \( x \) in the equation for \( \cos \theta \)

We know that \( x = \cos 40^\circ \). Therefore, we can substitute \( x \) into the equation for \( \cos \theta \):

\[ \cos \theta = 1 - 2(\cos 40^\circ)^2 \]

### Step 2: Use the double angle identity

Using the double angle identity for cosine, we know that:

\[ \cos 2A = 2\cos^2 A - 1 \]

Thus, we can rewrite \( 1 - 2(\cos 40^\circ)^2 \) as:

\[ \cos \theta = -\cos 80^\circ \]

### Step 3: Set up the equation for \( \theta \)

Now we have:

\[ \cos \theta = -\cos 80^\circ \]

This implies:

\[ \cos \theta = \cos(180^\circ - 80^\circ) \quad \text{or} \quad \cos \theta = \cos(180^\circ + 80^\circ) \]

### Step 4: Solve for \( \theta \)

From the above, we can find the possible values of \( \theta \):

1. \( \theta = 180^\circ - 80^\circ = 100^\circ \)
2. \( \theta = 180^\circ + 80^\circ = 260^\circ \)

### Step 5: Conclusion

Thus, the possible values of \( \theta \) lying between \( 0^\circ \) and \( 360^\circ \) are:

\[ \theta = 100^\circ \quad \text{and} \quad \theta = 260^\circ \]

### Final Answer:

The possible values of \( \theta \) are \( 100^\circ \) and \( 260^\circ \).

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