Trigonometry Examples

$\sqrt{2} \sin (x) \sec (x) = 2 \sin (x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\sqrt{2} \sin (x) \sec (x)$ .

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Step 1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 1.1.2

Multiply $\sqrt{2} \sin (x) \frac{1}{\cos (x)}$ .

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Step 1.1.2.1

Combine $\frac{1}{\cos (x)}$ and $\sqrt{2}$ .

$\frac{\sqrt{2}}{\cos (x)} \sin (x) = 2 \sin (x)$

Step 1.1.2.2

Combine $\frac{\sqrt{2}}{\cos (x)}$ and $\sin (x)$ .

$\frac{\sqrt{2} \sin (x)}{\cos (x)} = 2 \sin (x)$

Step 2

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) \frac{\sqrt{2} \sin (x)}{\cos (x)} = \cos (x) (2 \sin (x))$

Step 3

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

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Step 3.1

Cancel the common factor.

$\cos (x) \frac{\sqrt{2} \sin (x)}{\cos (x)} = \cos (x) (2 \sin (x))$

Step 3.2

Rewrite the expression.

$\sqrt{2} \sin (x) = \cos (x) (2 \sin (x))$

Step 4

Rewrite using the commutative property of multiplication.

$\sqrt{2} \sin (x) = 2 \cos (x) \sin (x)$

Step 5

Reorder $2 \cos (x)$ and $\sin (x)$ .

$\sqrt{2} \sin (x) = \sin (x) (2 \cos (x))$

Step 6

Reorder $\sin (x)$ and $2$ .

$\sqrt{2} \sin (x) = 2 \cdot \sin (x) \cos (x)$

Step 7

Apply the sine double-angle identity.

$\sqrt{2} \sin (x) = \sin (2 x)$

Step 8

Subtract $\sin (2 x)$ from both sides of the equation.

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

Step 9

Simplify each term.

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Step 9.1

Apply the sine double-angle identity.

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

Step 9.2

Multiply $2$ by $- 1$ .

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 11

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

$\sin (x) = 0$

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

Step 12

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

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Step 12.1

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

$\sin (x) = 0$

Step 12.2

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

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Step 12.2.1

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

$x = \arcsin (0)$

Step 12.2.2

Simplify the right side.

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Step 12.2.2.1

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

$x = 0$

Step 12.2.3

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.

$x = π - 0$

Step 12.2.4

Subtract $0$ from $π$ .

$x = π$

Step 12.2.5

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

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Step 12.2.5.1

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

$\frac{2 π}{| b |}$

Step 12.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 12.2.5.3

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

$\frac{2 π}{1}$

Step 12.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 12.2.6

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

$x = 2 π n, π + 2 π n$ , for any integer $n$

Step 13

Set $\sqrt{2} - 2 \cos (x)$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $\sqrt{2} - 2 \cos (x)$ equal to $0$ .

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

Step 13.2

Solve $\sqrt{2} - 2 \cos (x) = 0$ for $x$ .

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Step 13.2.1

Subtract $\sqrt{2}$ from both sides of the equation.

$- 2 \cos (x) = - \sqrt{2}$

Step 13.2.2

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

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Step 13.2.2.1

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

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

Step 13.2.2.2

Simplify the left side.

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Step 13.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 13.2.2.2.1.1

Cancel the common factor.

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

Step 13.2.2.2.1.2

Divide $\cos (x)$ by $1$ .

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

Step 13.2.2.3

Simplify the right side.

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Step 13.2.2.3.1

Dividing two negative values results in a positive value.

$\cos (x) = \frac{\sqrt{2}}{2}$

Step 13.2.3

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

$x = \arccos (\frac{\sqrt{2}}{2})$

Step 13.2.4

Simplify the right side.

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Step 13.2.4.1

The exact value of $\arccos (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 13.2.5

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.

$x = 2 π - \frac{π}{4}$

Step 13.2.6

Simplify $2 π - \frac{π}{4}$ .

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Step 13.2.6.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 13.2.6.2

Combine fractions.

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Step 13.2.6.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 13.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 13.2.6.3

Simplify the numerator.

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Step 13.2.6.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 13.2.6.3.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 13.2.7

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

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Step 13.2.7.1

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

$\frac{2 π}{| b |}$

Step 13.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 13.2.7.3

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

$\frac{2 π}{1}$

Step 13.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 13.2.8

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

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 14

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

$x = 2 π n, π + 2 π n, \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 15

Consolidate $2 π n$ and $π + 2 π n$ to $π n$ .

$x = π n, \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$