Trigonometry Examples

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

Step 1

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

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

Step 2

Solve for $x$ .

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Step 2.1

Simplify the left side of the equation.

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Step 2.1.1

Apply pythagorean identity.

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

Step 2.1.2

Simplify with factoring out.

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Step 2.1.2.1

Move $- 1$ .

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

Step 2.1.2.5

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

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

Step 2.1.2.6

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

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

Step 2.1.3

Apply pythagorean identity.

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

Step 2.2

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

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Step 2.2.1

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

$\sin (2 x) (- \sin (2 x)) + \sin (2 x) = 0$

Step 2.2.2

Raise $\sin (2 x)$ to the power of $1$ .

$\sin (2 x) (- \sin (2 x)) + \sin (2 x) = 0$

Step 2.2.3

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

$\sin (2 x) (- \sin (2 x)) + \sin (2 x) \cdot 1 = 0$

Step 2.2.4

Factor $\sin (2 x)$ out of $\sin (2 x) (- \sin (2 x)) + \sin (2 x) \cdot 1$ .

$\sin (2 x) (- \sin (2 x) + 1) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\sin (2 x) = 0$

$- \sin (2 x) + 1 = 0$

Step 2.4

Set $\sin (2 x)$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $\sin (2 x)$ equal to $0$ .

$\sin (2 x) = 0$

Step 2.4.2

Solve $\sin (2 x) = 0$ for $x$ .

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Step 2.4.2.1

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

$2 x = \arcsin (0)$

Step 2.4.2.2

Simplify the right side.

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Step 2.4.2.2.1

The exact value of $\arcsin (0)$ is $0$ .

$2 x = 0$

Step 2.4.2.3

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 2.4.2.3.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.4.2.3.2

Simplify the left side.

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Step 2.4.2.3.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 2.4.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 2.4.2.3.3

Simplify the right side.

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Step 2.4.2.3.3.1

Divide $0$ by $2$ .

$x = 0$

Step 2.4.2.4

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

$2 x = π - 0$

Step 2.4.2.5

Solve for $x$ .

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Step 2.4.2.5.1

Simplify.

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Step 2.4.2.5.1.1

Multiply $- 1$ by $0$ .

$2 x = π + 0$

Step 2.4.2.5.1.2

Add $π$ and $0$ .

$2 x = π$

Step 2.4.2.5.2

Divide each term in $2 x = π$ by $2$ and simplify.

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Step 2.4.2.5.2.1

Divide each term in $2 x = π$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2}$

Step 2.4.2.5.2.2

Simplify the left side.

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Step 2.4.2.5.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2}$

Step 2.4.2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{2}$

Step 2.4.2.6

Find the period of $\sin (2 x)$ .

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Step 2.4.2.6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 2.4.2.6.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 2.4.2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 2.4.2.6.4

Cancel the common factor of $2$ .

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Step 2.4.2.6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 2.4.2.6.4.2

Divide $π$ by $1$ .

$π$

Step 2.4.2.7

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

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

Step 2.5

Set $- \sin (2 x) + 1$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $- \sin (2 x) + 1$ equal to $0$ .

$- \sin (2 x) + 1 = 0$

Step 2.5.2

Solve $- \sin (2 x) + 1 = 0$ for $x$ .

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Step 2.5.2.1

Subtract $1$ from both sides of the equation.

$- \sin (2 x) = - 1$

Step 2.5.2.2

Divide each term in $- \sin (2 x) = - 1$ by $- 1$ and simplify.

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Step 2.5.2.2.1

Divide each term in $- \sin (2 x) = - 1$ by $- 1$ .

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

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.5.2.2.2.2

Divide $\sin (2 x)$ by $1$ .

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

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Divide $- 1$ by $- 1$ .

$\sin (2 x) = 1$

Step 2.5.2.3

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

$2 x = \arcsin (1)$

Step 2.5.2.4

Simplify the right side.

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Step 2.5.2.4.1

The exact value of $\arcsin (1)$ is $\frac{π}{2}$ .

$2 x = \frac{π}{2}$

Step 2.5.2.5

Divide each term in $2 x = \frac{π}{2}$ by $2$ and simplify.

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Step 2.5.2.5.1

Divide each term in $2 x = \frac{π}{2}$ by $2$ .

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

Step 2.5.2.5.2

Simplify the left side.

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Step 2.5.2.5.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.5.2.1.1

Cancel the common factor.

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

Step 2.5.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 2.5.2.5.3

Simplify the right side.

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Step 2.5.2.5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 2.5.2.5.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 2.5.2.5.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

$x = \frac{π}{2 \cdot 2}$

Step 2.5.2.5.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 2.5.2.6

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

$2 x = π - \frac{π}{2}$

Step 2.5.2.7

Solve for $x$ .

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Step 2.5.2.7.1

Simplify.

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Step 2.5.2.7.1.1

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

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

Step 2.5.2.7.1.2

Combine $π$ and $\frac{2}{2}$ .

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

Step 2.5.2.7.1.3

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 2 - π}{2}$

Step 2.5.2.7.1.4

Subtract $π$ from $π \cdot 2$ .

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Step 2.5.2.7.1.4.1

Reorder $π$ and $2$ .

$2 x = \frac{2 \cdot π - π}{2}$

Step 2.5.2.7.1.4.2

Subtract $π$ from $2 \cdot π$ .

$2 x = \frac{π}{2}$

Step 2.5.2.7.2

Divide each term in $2 x = \frac{π}{2}$ by $2$ and simplify.

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Step 2.5.2.7.2.1

Divide each term in $2 x = \frac{π}{2}$ by $2$ .

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

Step 2.5.2.7.2.2

Simplify the left side.

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Step 2.5.2.7.2.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.7.2.2.1.1

Cancel the common factor.

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

Step 2.5.2.7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 2.5.2.7.2.3

Simplify the right side.

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Step 2.5.2.7.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 2.5.2.7.2.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 2.5.2.7.2.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

$x = \frac{π}{2 \cdot 2}$

Step 2.5.2.7.2.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 2.5.2.8

Find the period of $\sin (2 x)$ .

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Step 2.5.2.8.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 2.5.2.8.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 2.5.2.8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 2.5.2.8.4

Cancel the common factor of $2$ .

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Step 2.5.2.8.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 2.5.2.8.4.2

Divide $π$ by $1$ .

$π$

Step 2.5.2.9

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

$x = \frac{π}{4} + π n$ , for any integer $n$

Step 2.6

The final solution is all the values that make $\sin (2 x) (- \sin (2 x) + 1) = 0$ true.

$x = π n, \frac{π}{2} + π n, \frac{π}{4} + π n$ , for any integer $n$

Step 3

Consolidate $π n$ and $\frac{π}{2} + π n$ to $\frac{π n}{2}$ .

$x = \frac{π n}{2}, \frac{π}{4} + π n$ , for any integer $n$