Trigonometry Examples

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

Step 1

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

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

Step 2

Separate fractions.

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

Step 3

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

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

Step 4

Divide $\frac{\sin (x)}{\cos (x)}$ by $1$ .

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

Step 5

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

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

Step 6

Separate fractions.

$\tan (x) \sec (x) = \frac{- \sqrt{2}}{1} \cdot \frac{\sin (x)}{\cos (x)}$

Step 7

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

$\tan (x) \sec (x) = \frac{- \sqrt{2}}{1} \cdot \tan (x)$

Step 8

Divide $- \sqrt{2}$ by $1$ .

$\tan (x) \sec (x) = - \sqrt{2} \tan (x)$

Step 9

Add $\sqrt{2} \tan (x)$ to both sides of the equation.

$\tan (x) \sec (x) + \sqrt{2} \tan (x) = 0$

Step 10

Factor $\tan (x)$ out of $\tan (x) \sec (x) + \sqrt{2} \tan (x)$ .

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Step 10.1

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

$\tan (x) \sec (x) + \tan (x) \sqrt{2} = 0$

Step 10.2

Factor $\tan (x)$ out of $\tan (x) \sec (x) + \tan (x) \sqrt{2}$ .

$\tan (x) (\sec (x) + \sqrt{2}) = 0$

Step 11

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

$\tan (x) = 0$

$\sec (x) + \sqrt{2} = 0$

Step 12

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

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Step 12.1

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

$\tan (x) = 0$

Step 12.2

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

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Step 12.2.1

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

$x = \arctan (0)$

Step 12.2.2

Simplify the right side.

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Step 12.2.2.1

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

$x = 0$

Step 12.2.3

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.

$x = π + 0$

Step 12.2.4

Add $π$ and $0$ .

$x = π$

Step 12.2.5

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

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Step 12.2.5.1

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

$\frac{π}{| b |}$

Step 12.2.5.2

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

$\frac{π}{| 1 |}$

Step 12.2.5.3

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

$\frac{π}{1}$

Step 12.2.5.4

Divide $π$ by $1$ .

$π$

Step 12.2.6

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

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

Step 13

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

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Step 13.1

Set $\sec (x) + \sqrt{2}$ equal to $0$ .

$\sec (x) + \sqrt{2} = 0$

Step 13.2

Solve $\sec (x) + \sqrt{2} = 0$ for $x$ .

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Step 13.2.1

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

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

Step 13.2.2

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

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

Step 13.2.3

Simplify the right side.

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Step 13.2.3.1

The exact value of $arcsec (- \sqrt{2})$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 13.2.4

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{3 π}{4}$

Step 13.2.5

Simplify $2 π - \frac{3 π}{4}$ .

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Step 13.2.5.1

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

$x = 2 π \cdot \frac{4}{4} - \frac{3 π}{4}$

Step 13.2.5.2

Combine fractions.

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Step 13.2.5.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{3 π}{4}$

Step 13.2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Step 13.2.5.3

Simplify the numerator.

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Step 13.2.5.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - 3 π}{4}$

Step 13.2.5.3.2

Subtract $3 π$ from $8 π$ .

$x = \frac{5 π}{4}$

Step 13.2.6

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

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Step 13.2.6.1

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

$\frac{2 π}{| b |}$

Step 13.2.6.2

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

$\frac{2 π}{| 1 |}$

Step 13.2.6.3

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

$\frac{2 π}{1}$

Step 13.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 13.2.7

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

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

Step 14

The final solution is all the values that make $\tan (x) (\sec (x) + \sqrt{2}) = 0$ true.

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

Step 15

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

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