Trigonometry Examples

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

Step 1

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

$\frac{\sqrt{2} \sin (x)}{\cos (x)} + \frac{\sqrt{2} \cos (x)}{\cos (x)} > \frac{0}{\cos (x)}$

Step 2

Separate fractions.

$\frac{\sqrt{2}}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sqrt{2} \cos (x)}{\cos (x)} > \frac{0}{\cos (x)}$

Step 3

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

$\frac{\sqrt{2}}{1} \cdot \tan (x) + \frac{\sqrt{2} \cos (x)}{\cos (x)} > \frac{0}{\cos (x)}$

Step 4

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

$\sqrt{2} \tan (x) + \frac{\sqrt{2} \cos (x)}{\cos (x)} > \frac{0}{\cos (x)}$

Step 5

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

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Step 5.1

Cancel the common factor.

$\sqrt{2} \tan (x) + \frac{\sqrt{2} \cos (x)}{\cos (x)} > \frac{0}{\cos (x)}$

Step 5.2

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

$\sqrt{2} \tan (x) + \sqrt{2} > \frac{0}{\cos (x)}$

Step 6

Separate fractions.

$\sqrt{2} \tan (x) + \sqrt{2} > \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 7

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

$\sqrt{2} \tan (x) + \sqrt{2} > \frac{0}{1} \cdot \sec (x)$

Step 8

Divide $0$ by $1$ .

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

Step 9

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

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

Step 10

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

$\sqrt{2} \tan (x) > - \sqrt{2}$

Step 11

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

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Step 11.1

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

$\frac{\sqrt{2} \tan (x)}{\sqrt{2}} > \frac{- \sqrt{2}}{\sqrt{2}}$

Step 11.2

Simplify the left side.

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Step 11.2.1

Cancel the common factor of $\sqrt{2}$ .

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Step 11.2.1.1

Cancel the common factor.

$\frac{\sqrt{2} \tan (x)}{\sqrt{2}} > \frac{- \sqrt{2}}{\sqrt{2}}$

Step 11.2.1.2

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

$\tan (x) > \frac{- \sqrt{2}}{\sqrt{2}}$

Step 11.3

Simplify the right side.

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Step 11.3.1

Cancel the common factor of $\sqrt{2}$ .

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Step 11.3.1.1

Cancel the common factor.

$\tan (x) > \frac{- \sqrt{2}}{\sqrt{2}}$

Step 11.3.1.2

Divide $- 1$ by $1$ .

$\tan (x) > - 1$

Step 12

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

$x > \arctan (- 1)$

Step 13

Simplify the right side.

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Step 13.1

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

$x > - \frac{π}{4}$

Step 14

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

$x = - \frac{π}{4} - π$

Step 15

Simplify the expression to find the second solution.

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Step 15.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 15.2

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$x = \frac{3 π}{4}$

Step 16

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

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Step 16.1

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

$\frac{π}{| b |}$

Step 16.2

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

$\frac{π}{| 1 |}$

Step 16.3

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

$\frac{π}{1}$

Step 16.4

Divide $π$ by $1$ .

$π$

Step 17

Add $π$ to every negative angle to get positive angles.

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Step 17.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 17.2

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

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 17.3

Combine fractions.

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Step 17.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 17.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 17.4

Simplify the numerator.

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Step 17.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 17.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 17.5

List the new angles.

$x = \frac{3 π}{4}$

Step 18

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

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

Step 19

Use each root to create test intervals.

$\frac{3 π}{4} < x < \frac{7 π}{4}$

Step 20

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 20.1

Test a value on the interval $\frac{3 π}{4} < x < \frac{7 π}{4}$ to see if it makes the inequality true.

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Step 20.1.1

Choose a value on the interval $\frac{3 π}{4} < x < \frac{7 π}{4}$ and see if this value makes the original inequality true.

$x = 4$

Step 20.1.2

Replace $x$ with $4$ in the original inequality.

$\sqrt{2} \sin (4) + \sqrt{2} \cos (4) > 0$

Step 20.1.3

The left side $- 1.99467202$ is not greater than the right side $0$ , which means that the given statement is false.

False

Step 20.2

Compare the intervals to determine which ones satisfy the original inequality.

$\frac{3 π}{4} < x < \frac{7 π}{4}$ False

Step 21

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution