Trigonometry Examples

$\tan (h (- \sqrt{3}))$

Step 1

Find the asymptotes.

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Step 1.1

For any $y = \tan (x)$ , vertical asymptotes occur at $x = \frac{π}{2} + n π$ , where $n$ is an integer. Use the basic period for $y = \tan (x)$ , $(- \frac{π}{2}, \frac{π}{2})$ , to find the vertical asymptotes for $y = \tan (- x \sqrt{3})$ . Set the inside of the tangent function, $b x + c$ , for $y = a \tan (b x + c) + d$ equal to $- \frac{π}{2}$ to find where the vertical asymptote occurs for $y = \tan (- x \sqrt{3})$ .

$- x \sqrt{3} = - \frac{π}{2}$

Step 1.2

Divide each term in $- x \sqrt{3} = - \frac{π}{2}$ by $- \sqrt{3}$ and simplify.

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Step 1.2.1

Divide each term in $- x \sqrt{3} = - \frac{π}{2}$ by $- \sqrt{3}$ .

$\frac{- x \sqrt{3}}{- \sqrt{3}} = \frac{- \frac{π}{2}}{- \sqrt{3}}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Dividing two negative values results in a positive value.

$\frac{x \sqrt{3}}{\sqrt{3}} = \frac{- \frac{π}{2}}{- \sqrt{3}}$

Step 1.2.2.2

Cancel the common factor of $\sqrt{3}$ .

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Step 1.2.2.2.1

Cancel the common factor.

$\frac{x \sqrt{3}}{\sqrt{3}} = \frac{- \frac{π}{2}}{- \sqrt{3}}$

Step 1.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- \frac{π}{2}}{- \sqrt{3}}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{\frac{π}{2}}{\sqrt{3}}$

Step 1.2.3.2

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{\sqrt{3}}$

Step 1.2.3.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.2.3.4

Combine and simplify the denominator.

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Step 1.2.3.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \frac{π}{2} \cdot \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 1.2.3.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{π}{2} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 1.2.3.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{π}{2} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 1.2.3.4.4

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

$x = \frac{π}{2} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 1.2.3.4.5

Add $1$ and $1$ .

$x = \frac{π}{2} \cdot \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 1.2.3.4.6

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

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Step 1.2.3.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 1.2.3.4.6.2

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

$x = \frac{π}{2} \cdot \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 1.2.3.4.6.3

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

$x = \frac{π}{2} \cdot \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.2.3.4.6.4

Cancel the common factor of $2$ .

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Step 1.2.3.4.6.4.1

Cancel the common factor.

$x = \frac{π}{2} \cdot \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.2.3.4.6.4.2

Rewrite the expression.

$x = \frac{π}{2} \cdot \frac{\sqrt{3}}{3^{1}}$

Step 1.2.3.4.6.5

Evaluate the exponent.

$x = \frac{π}{2} \cdot \frac{\sqrt{3}}{3}$

Step 1.2.3.5

Multiply $\frac{π}{2} \cdot \frac{\sqrt{3}}{3}$ .

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Step 1.2.3.5.1

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

$x = \frac{π \sqrt{3}}{2 \cdot 3}$

Step 1.2.3.5.2

Multiply $2$ by $3$ .

$x = \frac{π \sqrt{3}}{6}$

Step 1.3

Set the inside of the tangent function $- x \sqrt{3}$ equal to $\frac{π}{2}$ .

$- x \sqrt{3} = \frac{π}{2}$

Step 1.4

Divide each term in $- x \sqrt{3} = \frac{π}{2}$ by $- \sqrt{3}$ and simplify.

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Step 1.4.1

Divide each term in $- x \sqrt{3} = \frac{π}{2}$ by $- \sqrt{3}$ .

$\frac{- x \sqrt{3}}{- \sqrt{3}} = \frac{\frac{π}{2}}{- \sqrt{3}}$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Dividing two negative values results in a positive value.

$\frac{x \sqrt{3}}{\sqrt{3}} = \frac{\frac{π}{2}}{- \sqrt{3}}$

Step 1.4.2.2

Cancel the common factor of $\sqrt{3}$ .

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Step 1.4.2.2.1

Cancel the common factor.

$\frac{x \sqrt{3}}{\sqrt{3}} = \frac{\frac{π}{2}}{- \sqrt{3}}$

Step 1.4.2.2.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{- \sqrt{3}}$

Step 1.4.3

Simplify the right side.

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Step 1.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{- \sqrt{3}}$

Step 1.4.3.2

Cancel the common factor of $1$ and $- 1$ .

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Step 1.4.3.2.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 1.4.3.2.2

Move the negative in front of the fraction.

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

Step 1.4.3.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.4.3.4

Combine and simplify the denominator.

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Step 1.4.3.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.4.3.4.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 1.4.3.4.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 1.4.3.4.4

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

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

Step 1.4.3.4.5

Add $1$ and $1$ .

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

Step 1.4.3.4.6

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

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Step 1.4.3.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 1.4.3.4.6.2

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

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

Step 1.4.3.4.6.3

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

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

Step 1.4.3.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.3.4.6.4.1

Cancel the common factor.

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

Step 1.4.3.4.6.4.2

Rewrite the expression.

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

Step 1.4.3.4.6.5

Evaluate the exponent.

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

Step 1.4.3.5

Multiply $\frac{π}{2} (- \frac{\sqrt{3}}{3})$ .

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Step 1.4.3.5.1

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

$x = - \frac{π \sqrt{3}}{2 \cdot 3}$

Step 1.4.3.5.2

Multiply $2$ by $3$ .

$x = - \frac{π \sqrt{3}}{6}$

Step 1.5

The basic period for $y = \tan (- x \sqrt{3})$ will occur at $(\frac{π \sqrt{3}}{6}, - \frac{π \sqrt{3}}{6})$ , where $\frac{π \sqrt{3}}{6}$ and $- \frac{π \sqrt{3}}{6}$ are vertical asymptotes.

$(\frac{π \sqrt{3}}{6}, - \frac{π \sqrt{3}}{6})$

Step 1.6

Find the period $\frac{π}{| b |}$ to find where the vertical asymptotes exist.

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Step 1.6.1

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

$\frac{π}{1}$

Step 1.6.2

Divide $π$ by $1$ .

$π$

Step 2

Use the form $a \tan (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 1$

$b = \sqrt{3} \cdot - 1$

$c = 0$

$d = 0$

Step 3

Since the graph of the function $t a n$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Step 4

Find the period of $\tan (- x \sqrt{3})$ .

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Step 4.1

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

$\frac{π}{| b |}$

Step 4.2

Replace $b$ with $\sqrt{3} \cdot - 1$ in the formula for period.

$\frac{π}{| \sqrt{3} \cdot - 1 |}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Move $- 1$ to the left of $\sqrt{3}$ .

$\frac{π}{| - 1 \cdot \sqrt{3} |}$

Step 4.3.2

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

$\frac{π}{| - \sqrt{3} |}$

Step 4.3.3

$- \sqrt{3}$ is approximately $- 1.7320508$ which is negative so negate $- \sqrt{3}$ and remove the absolute value

$\frac{π}{\sqrt{3}}$

Step 4.4

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

$\frac{π}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

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

$\frac{π \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.5.2

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{π \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.5.3

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{π \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.5.4

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

$\frac{π \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.5.5

Add $1$ and $1$ .

$\frac{π \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.5.6

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

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Step 4.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{π \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.5.6.2

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

$\frac{π \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.5.6.3

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

$\frac{π \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.5.6.4

Cancel the common factor of $2$ .

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Step 4.5.6.4.1

Cancel the common factor.

$\frac{π \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.5.6.4.2

Rewrite the expression.

$\frac{π \sqrt{3}}{3^{1}}$

Step 4.5.6.5

Evaluate the exponent.

$\frac{π \sqrt{3}}{3}$

Step 5

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 5.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 5.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{0}{\sqrt{3} \cdot - 1}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Move $- 1$ to the left of $\sqrt{3}$ .

Phase Shift: $\frac{0}{- 1 \cdot \sqrt{3}}$

Step 5.3.2

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

Phase Shift: $\frac{0}{- \sqrt{3}}$

Step 5.4

Divide $0$ by $- \sqrt{3}$ .

Phase Shift: $0$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{π \sqrt{3}}{3}$

Phase Shift: None

Vertical Shift: None

Step 7

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

$π$

Amplitude: None

Period: $\frac{π \sqrt{3}}{3}$

Phase Shift: None

Vertical Shift: None

Step 8