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 the left side of the equation.

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Step 2.1

Simplify terms.

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the sine double-angle identity.

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

Step 2.1.1.2

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

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

Step 2.1.1.3

Apply the distributive property.

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

Step 2.1.1.4

Multiply $- 1$ by $1$ .

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

Step 2.1.1.5

Multiply $- 2$ by $- 1$ .

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

Step 2.1.2

Simplify with factoring out.

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Step 2.1.2.1

Subtract $1$ from $- 1$ .

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

Step 2.1.2.2

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

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

Step 2.1.2.3

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

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

Step 2.1.2.4

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

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

Step 2.2

Apply pythagorean identity.

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

Step 3

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

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Step 3.1

Factor $2 \cos (x)$ out of $2 \sin (x) \cos (x)$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

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

$\cos (x) = 0$

$\sin (x) - \cos (x) = 0$

Step 5

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

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Step 5.1

Set $\cos (x)$ equal to $0$ .

$\cos (x) = 0$

Step 5.2

Solve $\cos (x) = 0$ for $x$ .

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Step 5.2.1

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

$x = \arccos (0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 5.2.3

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 5.2.4

Simplify $2 π - \frac{π}{2}$ .

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Step 5.2.4.1

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 5.2.4.2

Combine fractions.

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Step 5.2.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 5.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 5.2.4.3

Simplify the numerator.

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Step 5.2.4.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 5.2.4.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 5.2.5

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

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Step 5.2.5.1

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

$\frac{2 π}{| b |}$

Step 5.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.5.3

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

$\frac{2 π}{1}$

Step 5.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.6

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 6

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

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Step 6.1

Set $\sin (x) - \cos (x)$ equal to $0$ .

$\sin (x) - \cos (x) = 0$

Step 6.2

Solve $\sin (x) - \cos (x) = 0$ for $x$ .

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Step 6.2.1

Divide each term in the equation by $\cos (x)$ .

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

Step 6.2.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) + \frac{- \cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 6.2.3

Cancel the common factor of $\cos (x)$ .

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Step 6.2.3.1

Cancel the common factor.

$\tan (x) + \frac{- \cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 6.2.3.2

Divide $- 1$ by $1$ .

$\tan (x) - 1 = \frac{0}{\cos (x)}$

Step 6.2.4

Separate fractions.

$\tan (x) - 1 = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 6.2.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\tan (x) - 1 = \frac{0}{1} \cdot \sec (x)$

Step 6.2.6

Divide $0$ by $1$ .

$\tan (x) - 1 = 0 \sec (x)$

Step 6.2.7

Multiply $0$ by $\sec (x)$ .

$\tan (x) - 1 = 0$

Step 6.2.8

Add $1$ to both sides of the equation.

$\tan (x) = 1$

Step 6.2.9

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

$x = \arctan (1)$

Step 6.2.10

Simplify the right side.

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Step 6.2.10.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 6.2.11

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + \frac{π}{4}$

Step 6.2.12

Simplify $π + \frac{π}{4}$ .

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Step 6.2.12.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 6.2.12.2

Combine fractions.

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Step 6.2.12.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 6.2.12.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 6.2.12.3

Simplify the numerator.

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Step 6.2.12.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 6.2.12.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 6.2.13

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

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Step 6.2.13.1

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

$\frac{π}{| b |}$

Step 6.2.13.2

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

$\frac{π}{| 1 |}$

Step 6.2.13.3

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

$\frac{π}{1}$

Step 6.2.13.4

Divide $π$ by $1$ .

$π$

Step 6.2.14

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

$x = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 7

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

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n, \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 8

Consolidate the answers.

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Step 8.1

Consolidate $\frac{π}{2} + 2 π n$ and $\frac{3 π}{2} + 2 π n$ to $\frac{π}{2} + π n$ .

$x = \frac{π}{2} + π n, \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 8.2

Consolidate $\frac{π}{4} + π n$ and $\frac{5 π}{4} + π n$ to $\frac{π}{4} + π n$ .

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