Trigonometry Examples

$\cos (h (\frac{\sqrt{3}}{2}))$

Step 1

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

$a = 1$

$b = \frac{\sqrt{3}}{2}$

$c = 0$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

Find the period of $\cos (\frac{x \sqrt{3}}{2})$ .

Tap for more steps...

Step 3.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $\frac{\sqrt{3}}{2}$ in the formula for period.

$\frac{2 π}{| \frac{\sqrt{3}}{2} |}$

Step 3.3

$\frac{\sqrt{3}}{2}$ is approximately $0.8660254$ which is positive so remove the absolute value

$\frac{2 π}{\frac{\sqrt{3}}{2}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{\sqrt{3}}$

Step 3.5

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

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

Step 3.6

Combine and simplify the denominator.

Tap for more steps...

Step 3.6.1

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

$2 π \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

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

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

Step 3.6.5

Add $1$ and $1$ .

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

Step 3.6.6

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

Tap for more steps...

Step 3.6.6.1

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

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

Step 3.6.6.2

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

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

Step 3.6.6.3

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

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

Step 3.6.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.6.6.4.1

Cancel the common factor.

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

Step 3.6.6.4.2

Rewrite the expression.

$2 π \frac{2 \sqrt{3}}{3^{1}}$

Step 3.6.6.5

Evaluate the exponent.

$2 π \frac{2 \sqrt{3}}{3}$

Step 3.7

Multiply $2 π \frac{2 \sqrt{3}}{3}$ .

Tap for more steps...

Step 3.7.1

Combine $\frac{2 \sqrt{3}}{3}$ and $2$ .

$\frac{2 \sqrt{3} \cdot 2}{3} π$

Step 3.7.2

Multiply $2$ by $2$ .

$\frac{4 \sqrt{3}}{3} π$

Step 3.7.3

Combine $\frac{4 \sqrt{3}}{3}$ and $π$ .

$\frac{4 \sqrt{3} π}{3}$

Step 4

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

Tap for more steps...

Step 4.1

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

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

Step 4.2

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

Phase Shift: $\frac{0}{\frac{\sqrt{3}}{2}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{2}{\sqrt{3}})$

Step 4.4

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

Phase Shift: $0 (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 4.5

Combine and simplify the denominator.

Tap for more steps...

Step 4.5.1

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

Phase Shift: $0 (\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})$

Step 4.5.2

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

Phase Shift: $0 (\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})$

Step 4.5.3

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

Phase Shift: $0 (\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})$

Step 4.5.4

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

Phase Shift: $0 (\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}})$

Step 4.5.5

Add $1$ and $1$ .

Phase Shift: $0 (\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}})$

Step 4.5.6

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

Tap for more steps...

Step 4.5.6.1

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

Phase Shift: $0 (\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})$

Step 4.5.6.2

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

Phase Shift: $0 (\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})$

Step 4.5.6.3

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

Phase Shift: $0 (\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})$

Step 4.5.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.5.6.4.1

Cancel the common factor.

Phase Shift: $0 (\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})$

Step 4.5.6.4.2

Rewrite the expression.

Phase Shift: $0 (\frac{2 \sqrt{3}}{3})$

Step 4.5.6.5

Evaluate the exponent.

Phase Shift: $0 (\frac{2 \sqrt{3}}{3})$

Step 4.6

Multiply $0$ by $\frac{2 \sqrt{3}}{3}$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{4 \sqrt{3} π}{3}$

Phase Shift: None

Vertical Shift: None

Step 6

Select a few points to graph.

Tap for more steps...

Step 6.1

Find the point at $x = 0$ .

Tap for more steps...

Step 6.1.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \cos (\frac{(0) \sqrt{3}}{2})$

Step 6.1.2

Simplify the result.

Tap for more steps...

Step 6.1.2.1

Cancel the common factor of $0$ and $2$ .

Tap for more steps...

Step 6.1.2.1.1

Factor $2$ out of $(0) \sqrt{3}$ .

$f (0) = \cos (\frac{2 ((0) \sqrt{3})}{2})$

Step 6.1.2.1.2

Cancel the common factors.

Tap for more steps...

Step 6.1.2.1.2.1

Factor $2$ out of $2$ .

$f (0) = \cos (\frac{2 ((0) \sqrt{3})}{2 (1)})$

Step 6.1.2.1.2.2

Cancel the common factor.

$f (0) = \cos (\frac{2 ((0) \sqrt{3})}{2 \cdot 1})$

Step 6.1.2.1.2.3

Rewrite the expression.

$f (0) = \cos (\frac{(0) \sqrt{3}}{1})$

Step 6.1.2.1.2.4

Divide $(0) \sqrt{3}$ by $1$ .

$f (0) = \cos ((0) \sqrt{3})$

Step 6.1.2.2

Multiply $0$ by $\sqrt{3}$ .

$f (0) = \cos (0)$

Step 6.1.2.3

The exact value of $\cos (0)$ is $1$ .

$f (0) = 1$

Step 6.1.2.4

The final answer is $1$ .

$1$

Step 6.2

Find the point at $x = \frac{\sqrt{3} π}{3}$ .

Tap for more steps...

Step 6.2.1

Replace the variable $x$ with $\frac{\sqrt{3} π}{3}$ in the expression.

$f (\frac{\sqrt{3} π}{3}) = \cos (\frac{(\frac{\sqrt{3} π}{3}) \sqrt{3}}{2})$

Step 6.2.2

Simplify the result.

Tap for more steps...

Step 6.2.2.1

Combine $\frac{\sqrt{3} π}{3}$ and $\sqrt{3}$ .

$f (\frac{\sqrt{3} π}{3}) = \cos (\frac{\frac{\sqrt{3} π \sqrt{3}}{3}}{2})$

Step 6.2.2.2

Simplify the numerator.

Tap for more steps...

Step 6.2.2.2.1

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

$f (\frac{\sqrt{3} π}{3}) = \cos (\frac{\frac{\sqrt{3} \sqrt{3} π}{3}}{2})$

Step 6.2.2.2.2

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

$f (\frac{\sqrt{3} π}{3}) = \cos (\frac{\frac{\sqrt{3} \sqrt{3} π}{3}}{2})$

Step 6.2.2.2.3

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

$f (\frac{\sqrt{3} π}{3}) = \cos (\frac{\frac{{\sqrt{3}}^{1 + 1} π}{3}}{2})$

Step 6.2.2.2.4

Add $1$ and $1$ .

$f (\frac{\sqrt{3} π}{3}) = \cos (\frac{\frac{{\sqrt{3}}^{2} π}{3}}{2})$

Step 6.2.2.3

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

Tap for more steps...

Step 6.2.2.3.1

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

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

Step 6.2.2.3.2

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

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

Step 6.2.2.3.3

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

$f (\frac{\sqrt{3} π}{3}) = \cos (\frac{\frac{3^{\frac{2}{2}} π}{3}}{2})$

Step 6.2.2.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.2.2.3.4.1

Cancel the common factor.

$f (\frac{\sqrt{3} π}{3}) = \cos (\frac{\frac{3^{\frac{2}{2}} π}{3}}{2})$

Step 6.2.2.3.4.2

Rewrite the expression.

$f (\frac{\sqrt{3} π}{3}) = \cos (\frac{\frac{3 π}{3}}{2})$

Step 6.2.2.3.5

Evaluate the exponent.

$f (\frac{\sqrt{3} π}{3}) = \cos (\frac{\frac{3 π}{3}}{2})$

Step 6.2.2.4

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 6.2.2.4.1

Reduce the expression $\frac{3 π}{3}$ by cancelling the common factors.

Tap for more steps...

Step 6.2.2.4.1.1

Cancel the common factor.

$f (\frac{\sqrt{3} π}{3}) = \cos (\frac{\frac{3 π}{3}}{2})$

Step 6.2.2.4.1.2

Rewrite the expression.

$f (\frac{\sqrt{3} π}{3}) = \cos (\frac{\frac{π}{1}}{2})$

Step 6.2.2.4.2

Divide $π$ by $1$ .

$f (\frac{\sqrt{3} π}{3}) = \cos (\frac{π}{2})$

Step 6.2.2.5

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{\sqrt{3} π}{3}) = 0$

Step 6.2.2.6

The final answer is $0$ .

$0$

Step 6.3

Find the point at $x = \frac{2 \sqrt{3} π}{3}$ .

Tap for more steps...

Step 6.3.1

Replace the variable $x$ with $\frac{2 \sqrt{3} π}{3}$ in the expression.

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{(\frac{2 \sqrt{3} π}{3}) \sqrt{3}}{2})$

Step 6.3.2

Simplify the result.

Tap for more steps...

Step 6.3.2.1

Combine $\frac{2 \sqrt{3} π}{3}$ and $\sqrt{3}$ .

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{\frac{2 \sqrt{3} π \sqrt{3}}{3}}{2})$

Step 6.3.2.2

Simplify the numerator.

Tap for more steps...

Step 6.3.2.2.1

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

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{\frac{2 (\sqrt{3} \sqrt{3}) π}{3}}{2})$

Step 6.3.2.2.2

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

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{\frac{2 (\sqrt{3} \sqrt{3}) π}{3}}{2})$

Step 6.3.2.2.3

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

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{\frac{2 {\sqrt{3}}^{1 + 1} π}{3}}{2})$

Step 6.3.2.2.4

Add $1$ and $1$ .

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{\frac{2 {\sqrt{3}}^{2} π}{3}}{2})$

Step 6.3.2.3

Simplify the numerator.

Tap for more steps...

Step 6.3.2.3.1

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

Tap for more steps...

Step 6.3.2.3.1.1

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

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

Step 6.3.2.3.1.2

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

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

Step 6.3.2.3.1.3

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

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{\frac{2 \cdot (3^{\frac{2}{2}} π)}{3}}{2})$

Step 6.3.2.3.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.2.3.1.4.1

Cancel the common factor.

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{\frac{2 \cdot (3^{\frac{2}{2}} π)}{3}}{2})$

Step 6.3.2.3.1.4.2

Rewrite the expression.

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{\frac{2 \cdot (3 π)}{3}}{2})$

Step 6.3.2.3.1.5

Evaluate the exponent.

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{\frac{2 \cdot (3 π)}{3}}{2})$

Step 6.3.2.3.2

Multiply $2$ by $3$ .

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{\frac{6 π}{3}}{2})$

Step 6.3.2.4

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 6.3.2.4.1

Reduce the expression $\frac{6 π}{3}$ by cancelling the common factors.

Tap for more steps...

Step 6.3.2.4.1.1

Factor $3$ out of $6 π$ .

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{\frac{3 (2 π)}{3}}{2})$

Step 6.3.2.4.1.2

Factor $3$ out of $3$ .

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{\frac{3 (2 π)}{3 (1)}}{2})$

Step 6.3.2.4.1.3

Cancel the common factor.

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

Step 6.3.2.4.1.4

Rewrite the expression.

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{\frac{2 π}{1}}{2})$

Step 6.3.2.4.2

Divide $2 π$ by $1$ .

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{2 π}{2})$

Step 6.3.2.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.2.5.1

Cancel the common factor.

$f (\frac{2 \sqrt{3} π}{3}) = \cos (\frac{2 π}{2})$

Step 6.3.2.5.2

Divide $π$ by $1$ .

$f (\frac{2 \sqrt{3} π}{3}) = \cos (π)$

Step 6.3.2.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$f (\frac{2 \sqrt{3} π}{3}) = - \cos (0)$

Step 6.3.2.7

The exact value of $\cos (0)$ is $1$ .

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

Step 6.3.2.8

Multiply $- 1$ by $1$ .

$f (\frac{2 \sqrt{3} π}{3}) = - 1$

Step 6.3.2.9

The final answer is $- 1$ .

$- 1$

Step 6.4

Find the point at $x = \sqrt{3} π$ .

Tap for more steps...

Step 6.4.1

Replace the variable $x$ with $\sqrt{3} π$ in the expression.

$f (\sqrt{3} π) = \cos (\frac{(\sqrt{3} π) \sqrt{3}}{2})$

Step 6.4.2

Simplify the result.

Tap for more steps...

Step 6.4.2.1

Simplify the numerator.

Tap for more steps...

Step 6.4.2.1.1

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

$f (\sqrt{3} π) = \cos (\frac{\sqrt{3} \sqrt{3} π}{2})$

Step 6.4.2.1.2

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

$f (\sqrt{3} π) = \cos (\frac{\sqrt{3} \sqrt{3} π}{2})$

Step 6.4.2.1.3

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

$f (\sqrt{3} π) = \cos (\frac{{\sqrt{3}}^{1 + 1} π}{2})$

Step 6.4.2.1.4

Add $1$ and $1$ .

$f (\sqrt{3} π) = \cos (\frac{{\sqrt{3}}^{2} π}{2})$

Step 6.4.2.2

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

Tap for more steps...

Step 6.4.2.2.1

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

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

Step 6.4.2.2.2

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

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

Step 6.4.2.2.3

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

$f (\sqrt{3} π) = \cos (\frac{3^{\frac{2}{2}} π}{2})$

Step 6.4.2.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.4.2.2.4.1

Cancel the common factor.

$f (\sqrt{3} π) = \cos (\frac{3^{\frac{2}{2}} π}{2})$

Step 6.4.2.2.4.2

Rewrite the expression.

$f (\sqrt{3} π) = \cos (\frac{3 π}{2})$

Step 6.4.2.2.5

Evaluate the exponent.

$f (\sqrt{3} π) = \cos (\frac{3 π}{2})$

Step 6.4.2.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$f (\sqrt{3} π) = \cos (\frac{π}{2})$

Step 6.4.2.4

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\sqrt{3} π) = 0$

Step 6.4.2.5

The final answer is $0$ .

$0$

Step 6.5

Find the point at $x = \frac{4 \sqrt{3} π}{3}$ .

Tap for more steps...

Step 6.5.1

Replace the variable $x$ with $\frac{4 \sqrt{3} π}{3}$ in the expression.

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{(\frac{4 \sqrt{3} π}{3}) \sqrt{3}}{2})$

Step 6.5.2

Simplify the result.

Tap for more steps...

Step 6.5.2.1

Combine $\frac{4 \sqrt{3} π}{3}$ and $\sqrt{3}$ .

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{4 \sqrt{3} π \sqrt{3}}{3}}{2})$

Step 6.5.2.2

Simplify the numerator.

Tap for more steps...

Step 6.5.2.2.1

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

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{4 (\sqrt{3} \sqrt{3}) π}{3}}{2})$

Step 6.5.2.2.2

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

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{4 (\sqrt{3} \sqrt{3}) π}{3}}{2})$

Step 6.5.2.2.3

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

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{4 {\sqrt{3}}^{1 + 1} π}{3}}{2})$

Step 6.5.2.2.4

Add $1$ and $1$ .

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{4 {\sqrt{3}}^{2} π}{3}}{2})$

Step 6.5.2.3

Simplify the numerator.

Tap for more steps...

Step 6.5.2.3.1

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

Tap for more steps...

Step 6.5.2.3.1.1

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

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{4 {(3^{\frac{1}{2}})}^{2} π}{3}}{2})$

Step 6.5.2.3.1.2

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

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{4 \cdot (3^{\frac{1}{2} \cdot 2} π)}{3}}{2})$

Step 6.5.2.3.1.3

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

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{4 \cdot (3^{\frac{2}{2}} π)}{3}}{2})$

Step 6.5.2.3.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.5.2.3.1.4.1

Cancel the common factor.

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{4 \cdot (3^{\frac{2}{2}} π)}{3}}{2})$

Step 6.5.2.3.1.4.2

Rewrite the expression.

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{4 \cdot (3 π)}{3}}{2})$

Step 6.5.2.3.1.5

Evaluate the exponent.

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{4 \cdot (3 π)}{3}}{2})$

Step 6.5.2.3.2

Multiply $4$ by $3$ .

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{12 π}{3}}{2})$

Step 6.5.2.4

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 6.5.2.4.1

Reduce the expression $\frac{12 π}{3}$ by cancelling the common factors.

Tap for more steps...

Step 6.5.2.4.1.1

Factor $3$ out of $12 π$ .

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{3 (4 π)}{3}}{2})$

Step 6.5.2.4.1.2

Factor $3$ out of $3$ .

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{3 (4 π)}{3 (1)}}{2})$

Step 6.5.2.4.1.3

Cancel the common factor.

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{3 (4 π)}{3 \cdot 1}}{2})$

Step 6.5.2.4.1.4

Rewrite the expression.

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{\frac{4 π}{1}}{2})$

Step 6.5.2.4.2

Divide $4 π$ by $1$ .

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{4 π}{2})$

Step 6.5.2.5

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 6.5.2.5.1

Factor $2$ out of $4 π$ .

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{2 (2 π)}{2})$

Step 6.5.2.5.2

Cancel the common factors.

Tap for more steps...

Step 6.5.2.5.2.1

Factor $2$ out of $2$ .

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{2 (2 π)}{2 (1)})$

Step 6.5.2.5.2.2

Cancel the common factor.

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{2 (2 π)}{2 \cdot 1})$

Step 6.5.2.5.2.3

Rewrite the expression.

$f (\frac{4 \sqrt{3} π}{3}) = \cos (\frac{2 π}{1})$

Step 6.5.2.5.2.4

Divide $2 π$ by $1$ .

$f (\frac{4 \sqrt{3} π}{3}) = \cos (2 π)$

Step 6.5.2.6

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (\frac{4 \sqrt{3} π}{3}) = \cos (0)$

Step 6.5.2.7

The exact value of $\cos (0)$ is $1$ .

$f (\frac{4 \sqrt{3} π}{3}) = 1$

Step 6.5.2.8

The final answer is $1$ .

$1$

Step 6.6

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 1 \\ \frac{\sqrt{3} π}{3} & 0 \\ \frac{2 \sqrt{3} π}{3} & - 1 \\ \sqrt{3} π & 0 \\ \frac{4 \sqrt{3} π}{3} & 1 \end{array}$

Step 7

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

Amplitude: $1$

Period: $\frac{4 \sqrt{3} π}{3}$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & 1 \\ \frac{\sqrt{3} π}{3} & 0 \\ \frac{2 \sqrt{3} π}{3} & - 1 \\ \sqrt{3} π & 0 \\ \frac{4 \sqrt{3} π}{3} & 1 \end{array}$

Step 8