Trigonometry Examples

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

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\cos^{2} (x)$ from both sides of the equation.

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

Step 1.2

Subtract $1$ from both sides of the equation.

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

Step 2

Simplify $\sin^{2} (x) - \cos^{2} (x) - 1$ .

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Step 2.1

Move $- 1$ .

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

Step 2.2

Reorder $\sin^{2} (x)$ and $- 1$ .

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

Step 2.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

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

Step 2.7

Apply pythagorean identity.

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

Step 2.8

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

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

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $- 2$ .

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Step 3.1.2.1.1

Cancel the common factor.

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

Step 3.1.2.1.2

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

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $0$ by $- 2$ .

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

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 $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 3.6

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{2}$

Step 3.7

Simplify $2 π - \frac{π}{2}$ .

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Step 3.7.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 3.7.2

Combine fractions.

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Step 3.7.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 3.7.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 3.7.3

Simplify the numerator.

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Step 3.7.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 3.7.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

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 $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.8.2

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

$\frac{2 π}{| 1 |}$

Step 3.8.3

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

$\frac{2 π}{1}$

Step 3.8.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.9

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 4

Consolidate the answers.

$x = \frac{π}{2} + π n$ , for any integer $n$