Algebra Examples

$\sin^{2} (θ) + 2 \cos^{2} (θ) = 4$

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 $θ$ .

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Step 3.1

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

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Step 3.1.1

Subtract $4$ from $1$ .

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

Step 3.1.2

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

$\cos^{2} (θ) - 3 = 0$

Step 3.2

Add $3$ to both sides of the equation.

$\cos^{2} (θ) = 3$

Step 3.3

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

$\cos (θ) = \pm \sqrt{3}$

Step 3.4

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

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Step 3.4.1

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

$\cos (θ) = \sqrt{3}$

Step 3.4.2

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

$\cos (θ) = - \sqrt{3}$

Step 3.4.3

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

$\cos (θ) = \sqrt{3}, - \sqrt{3}$

Step 3.5

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

$\cos (θ) = \sqrt{3}$

$\cos (θ) = - \sqrt{3}$

Step 3.6

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

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Step 3.6.1

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

Step 3.7

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

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Step 3.7.1

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