Trigonometry Examples

$2 \sin^{2} (x) - \cos^{2} (x) = 2$

Step 1

Subtract $2$ from both sides of the equation.

$2 \sin^{2} (x) - \cos^{2} (x) - 2 = 0$

Step 2

Simplify $2 \sin^{2} (x) - \cos^{2} (x) - 2$ .

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Step 2.1

Move $- 2$ .

$2 \sin^{2} (x) - 2 - \cos^{2} (x) = 0$

Step 2.2

Reorder $2 \sin^{2} (x)$ and $- 2$ .

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

Step 2.3

Factor $- 2$ out of $- 2$ .

$- 2 (1) + 2 \sin^{2} (x) - \cos^{2} (x) = 0$

Step 2.4

Factor $- 2$ out of $2 \sin^{2} (x)$ .

$- 2 (1) - 2 (- \sin^{2} (x)) - \cos^{2} (x) = 0$

Step 2.5

Factor $- 2$ out of $- 2 (1) - 2 (- \sin^{2} (x))$ .

$- 2 (1 - \sin^{2} (x)) - \cos^{2} (x) = 0$

Step 2.6

Apply pythagorean identity.

$- 2 \cos^{2} (x) - \cos^{2} (x) = 0$

Step 2.7

Subtract $\cos^{2} (x)$ from $- 2 \cos^{2} (x)$ .

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

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $- 3 \cos^{2} (x) = 0$ by $- 3$ and simplify.

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Step 3.1.1

Divide each term in $- 3 \cos^{2} (x) = 0$ by $- 3$ .

$\frac{- 3 \cos^{2} (x)}{- 3} = \frac{0}{- 3}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $- 3$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{- 3 \cos^{2} (x)}{- 3} = \frac{0}{- 3}$

Step 3.1.2.1.2

Divide $\cos^{2} (x)$ by $1$ .

$\cos^{2} (x) = \frac{0}{- 3}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $0$ by $- 3$ .

$\cos^{2} (x) = 0$

Step 3.2

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

$\cos (x) = \pm \sqrt{0}$

Step 3.3

Simplify $\pm \sqrt{0}$ .

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Step 3.3.1

Rewrite $0$ as $0^{2}$ .

$\cos (x) = \pm \sqrt{0^{2}}$

Step 3.3.2

Pull terms out from under the radical, assuming positive real numbers.

$\cos (x) = \pm 0$

Step 3.3.3

Plus or minus $0$ is $0$ .

$\cos (x) = 0$

Step 3.4

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (0)$

Step 3.5

Simplify the right side.

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Step 3.5.1

The exact value of $\arccos (0)$ is $90$ .

$x = 90$

Step 3.6

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $360$ to find the solution in the fourth quadrant.

$x = 360 - 90$

Step 3.7

Subtract $90$ from $360$ .

$x = 270$

Step 3.8

Find the period of $\cos (x)$ .

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Step 3.8.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 3.8.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 3.8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 3.8.4

Divide $360$ by $1$ .

$360$

Step 3.9

The period of the $\cos (x)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$x = 90 + 360 n, 270 + 360 n$ , for any integer $n$

Step 4

Consolidate the answers.

$x = 90 + 180 n$ , for any integer $n$