Trigonometry Examples

$\tan (x) = \sqrt{\frac{3}{2}}$

Step 1

Simplify $\sqrt{\frac{3}{2}}$ .

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Step 1.1

Rewrite $\sqrt{\frac{3}{2}}$ as $\frac{\sqrt{3}}{\sqrt{2}}$ .

$\tan (x) = \frac{\sqrt{3}}{\sqrt{2}}$

Step 1.2

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\tan (x) = \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\tan (x) = \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\tan (x) = \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\tan (x) = \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\tan (x) = \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.3.5

Add $1$ and $1$ .

$\tan (x) = \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\tan (x) = \frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\tan (x) = \frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\tan (x) = \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\tan (x) = \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.3.6.4.2

Rewrite the expression.

$\tan (x) = \frac{\sqrt{3} \sqrt{2}}{2^{1}}$

Step 1.3.6.5

Evaluate the exponent.

$\tan (x) = \frac{\sqrt{3} \sqrt{2}}{2}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Combine using the product rule for radicals.

$\tan (x) = \frac{\sqrt{3 \cdot 2}}{2}$

Step 1.4.2

Multiply $3$ by $2$ .

$\tan (x) = \frac{\sqrt{6}}{2}$

Step 2

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

$x = \arctan (\frac{\sqrt{6}}{2})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arctan (\frac{\sqrt{6}}{2})$ .

$x = 0.88607712$

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.

$x = (3.14159265) + 0.88607712$

Step 5

Solve for $x$ .

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Step 5.1

Remove parentheses.

$x = 3.14159265 + 0.88607712$

Step 5.2

Remove parentheses.

$x = (3.14159265) + 0.88607712$

Step 5.3

Add $3.14159265$ and $0.88607712$ .

$x = 4.02766977$

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.

$x = 0.88607712 + π n, 4.02766977 + π n$ , for any integer $n$

Step 8

Consolidate $0.88607712 + π n$ and $4.02766977 + π n$ to $0.88607712 + π n$ .

$x = 0.88607712 + π n$ , for any integer $n$