Trigonometry Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

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

Step 1.2.1.2

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

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

Step 2

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

$x > \arctan (\frac{\sqrt{2}}{2})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arctan (\frac{\sqrt{2}}{2})$ .

$x > 0.6154797$

Step 4

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.

$x = (3.14159265) + 0.6154797$

Step 5

Solve for $x$ .

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Step 5.1

Remove parentheses.

$x = 3.14159265 + 0.6154797$

Step 5.2

Remove parentheses.

$x = (3.14159265) + 0.6154797$

Step 5.3

Add $3.14159265$ and $0.6154797$ .

$x = 3.75707236$

Step 6

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

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Step 6.1

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

$\frac{π}{| b |}$

Step 6.2

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

$\frac{π}{| 1 |}$

Step 6.3

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

$\frac{π}{1}$

Step 6.4

Divide $π$ by $1$ .

$π$

Step 7

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

$x = 0.6154797 + π n, 3.75707236 + π n$ , for any integer $n$

Step 8

Consolidate $0.6154797 + π n$ and $3.75707236 + π n$ to $0.6154797 + π n$ .

$x = 0.6154797 + π n$ , for any integer $n$

Step 9

Find the domain of $2 \tan (x) - \sqrt{2}$ .

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Step 9.1

Set the argument in $\tan (x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

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

Step 9.2

The domain is all values of $x$ that make the expression defined.

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

Step 10

Use each root to create test intervals.

$0.6154797 < x < \frac{π}{2}$

$\frac{π}{2} < x < 3.75707236$

Step 11

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 11.1

Test a value on the interval $0.6154797 < x < \frac{π}{2}$ to see if it makes the inequality true.

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Step 11.1.1

Choose a value on the interval $0.6154797 < x < \frac{π}{2}$ and see if this value makes the original inequality true.

$x = 1$

Step 11.1.2

Replace $x$ with $1$ in the original inequality.

$2 \tan (1) > \sqrt{2}$

Step 11.1.3

The left side $3.11481544$ is greater than the right side $1.41421356$ , which means that the given statement is always true.

True

Step 11.2

Test a value on the interval $\frac{π}{2} < x < 3.75707236$ to see if it makes the inequality true.

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Step 11.2.1

Choose a value on the interval $\frac{π}{2} < x < 3.75707236$ and see if this value makes the original inequality true.

$x = 3$

Step 11.2.2

Replace $x$ with $3$ in the original inequality.

$2 \tan (3) > \sqrt{2}$

Step 11.2.3

The left side $- 0.28509308$ is not greater than the right side $1.41421356$ , which means that the given statement is false.

False

Step 11.3

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

$0.6154797 < x < \frac{π}{2}$ True

$\frac{π}{2} < x < 3.75707236$ False

$0.6154797 < x < \frac{π}{2}$ True

$\frac{π}{2} < x < 3.75707236$ False

Step 12

The solution consists of all of the true intervals.

$0.6154797 + π n < x < \frac{π}{2} + π n$ , for any integer $n$

Step 13