Find All Complex Solutions sin(theta)^2+2cos(theta)^2=4

$\sin^{2} (θ) + 2 \cos^{2} (θ) = 4$

Step 1

Subtract $4$ from both sides of the equation.

$\sin^{2} (θ) + 2 \cos^{2} (θ) - 4 = 0$

Step 2

Replace $\sin^{2} (θ)$ with $1 - \cos^{2} (θ)$ .

$(1 - \cos^{2} (θ)) + 2 \cos^{2} (θ) - 4 = 0$

Step 3

Solve for $θ$ .

Simplify $1 - \cos^{2} (θ) + 2 \cos^{2} (θ) - 4$ .

Subtract $4$ from $1$ .

$- \cos^{2} (θ) + 2 \cos^{2} (θ) - 3 = 0$

Add $- \cos^{2} (θ)$ and $2 \cos^{2} (θ)$ .

$\cos^{2} (θ) - 3 = 0$

Add $3$ to both sides of the equation.

$\cos^{2} (θ) = 3$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$\cos (θ) = \pm \sqrt{3}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$\cos (θ) = \sqrt{3}$

Next, use the negative value of the $\pm$ to find the second solution.

$\cos (θ) = - \sqrt{3}$

The complete solution is the result of both the positive and negative portions of the solution.

$\cos (θ) = \sqrt{3}, - \sqrt{3}$

Set up each of the solutions to solve for $θ$ .

$\cos (θ) = \sqrt{3}$

$\cos (θ) = - \sqrt{3}$

Solve for $θ$ in $\cos (θ) = \sqrt{3}$ .

The range of cosine is $- 1 \leq y \leq 1$ . Since $\sqrt{3}$ does not fall in this range, there is no solution.

No solution

Solve for $θ$ in $\cos (θ) = - \sqrt{3}$ .

The range of cosine is $- 1 \leq y \leq 1$ . Since $- \sqrt{3}$ does not fall in this range, there is no solution.

No solution