Trigonometry Examples

$\tan (\frac{x}{2}) = \frac{\sqrt{1 - \frac{3}{5}}}{1 + (- \frac{3}{5})}$

Step 1

Simplify $\frac{\sqrt{1 - \frac{3}{5}}}{1 - \frac{3}{5}}$ .

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Write $1$ as a fraction with a common denominator.

$\tan (\frac{x}{2}) = \frac{\sqrt{\frac{5}{5} - \frac{3}{5}}}{1 - \frac{3}{5}}$

Step 1.1.2

Combine the numerators over the common denominator.

$\tan (\frac{x}{2}) = \frac{\sqrt{\frac{5 - 3}{5}}}{1 - \frac{3}{5}}$

Step 1.1.3

Subtract $3$ from $5$ .

$\tan (\frac{x}{2}) = \frac{\sqrt{\frac{2}{5}}}{1 - \frac{3}{5}}$

Step 1.1.4

Rewrite $\sqrt{\frac{2}{5}}$ as $\frac{\sqrt{2}}{\sqrt{5}}$ .

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2}}{\sqrt{5}}}{1 - \frac{3}{5}}$

Step 1.1.5

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}}{1 - \frac{3}{5}}$

Step 1.1.6

Combine and simplify the denominator.

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Step 1.1.6.1

Multiply $\frac{\sqrt{2}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2} \sqrt{5}}{\sqrt{5} \sqrt{5}}}{1 - \frac{3}{5}}$

Step 1.1.6.2

Raise $\sqrt{5}$ to the power of $1$ .

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}}{1 - \frac{3}{5}}$

Step 1.1.6.3

Raise $\sqrt{5}$ to the power of $1$ .

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}}{1 - \frac{3}{5}}$

Step 1.1.6.4

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

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}}{1 - \frac{3}{5}}$

Step 1.1.6.5

Add $1$ and $1$ .

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2} \sqrt{5}}{{\sqrt{5}}^{2}}}{1 - \frac{3}{5}}$

Step 1.1.6.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.1.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}}{1 - \frac{3}{5}}$

Step 1.1.6.6.2

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

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}}{1 - \frac{3}{5}}$

Step 1.1.6.6.3

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

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}}{1 - \frac{3}{5}}$

Step 1.1.6.6.4

Cancel the common factor of $2$ .

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Step 1.1.6.6.4.1

Cancel the common factor.

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2} \sqrt{5}}{5^{\frac{2}{2}}}}{1 - \frac{3}{5}}$

Step 1.1.6.6.4.2

Rewrite the expression.

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2} \sqrt{5}}{5^{1}}}{1 - \frac{3}{5}}$

Step 1.1.6.6.5

Evaluate the exponent.

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2} \sqrt{5}}{5}}{1 - \frac{3}{5}}$

Step 1.1.7

Simplify the numerator.

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Step 1.1.7.1

Combine using the product rule for radicals.

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{2 \cdot 5}}{5}}{1 - \frac{3}{5}}$

Step 1.1.7.2

Multiply $2$ by $5$ .

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{10}}{5}}{1 - \frac{3}{5}}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{10}}{5}}{\frac{5}{5} - \frac{3}{5}}$

Step 1.2.2

Combine the numerators over the common denominator.

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{10}}{5}}{\frac{5 - 3}{5}}$

Step 1.2.3

Subtract $3$ from $5$ .

$\tan (\frac{x}{2}) = \frac{\frac{\sqrt{10}}{5}}{\frac{2}{5}}$

Step 1.3

Multiply the numerator by the reciprocal of the denominator.

$\tan (\frac{x}{2}) = \frac{\sqrt{10}}{5} \cdot \frac{5}{2}$

Step 1.4

Cancel the common factor of $5$ .

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Step 1.4.1

Cancel the common factor.

$\tan (\frac{x}{2}) = \frac{\sqrt{10}}{5} \cdot \frac{5}{2}$

Step 1.4.2

Rewrite the expression.

$\tan (\frac{x}{2}) = \sqrt{10} \frac{1}{2}$

Step 1.5

Combine $\sqrt{10}$ and $\frac{1}{2}$ .

$\tan (\frac{x}{2}) = \frac{\sqrt{10}}{2}$

Step 2

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

$\frac{x}{2} = \arctan (\frac{\sqrt{10}}{2})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arctan (\frac{\sqrt{10}}{2})$ .

$\frac{x}{2} = 1.00685368$

Step 4

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \cdot 1.00685368$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \cdot 1.00685368$

Step 5.1.1.2

Rewrite the expression.

$x = 2 \cdot 1.00685368$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $2$ by $1.00685368$ .

$x = 2.01370737$

Step 6

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.

$\frac{x}{2} = (3.14159265) + 1.00685368$

Step 7

Solve for $x$ .

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Step 7.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 ((3.14159265) + 1.00685368)$

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Cancel the common factor of $2$ .

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Step 7.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 ((3.14159265) + 1.00685368)$

Step 7.2.1.1.2

Rewrite the expression.

$x = 2 ((3.14159265) + 1.00685368)$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $2 ((3.14159265) + 1.00685368)$ .

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Step 7.2.2.1.1

Add $3.14159265$ and $1.00685368$ .

$x = 2 \cdot 4.14844633$

Step 7.2.2.1.2

Multiply $2$ by $4.14844633$ .

$x = 8.29689267$

Step 8

Find the period of $\tan (\frac{x}{2})$ .

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Step 8.1

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

$\frac{π}{| b |}$

Step 8.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{π}{| \frac{1}{2} |}$

Step 8.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{2}}$

Step 8.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 8.5

Move $2$ to the left of $π$ .

$2 π$

Step 9

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

$x = 2.01370737 + 2 π n, 8.29689267 + 2 π n$ , for any integer $n$

Step 10

Consolidate $2.01370737 + 2 π n$ and $8.29689267 + 2 π n$ to $2.01370737 + 2 π n$ .

$x = 2.01370737 + 2 π n$ , for any integer $n$