Trigonometry Examples

$\cos (x) = \frac{4}{3 \sqrt{3}}$

Step 1

Simplify $\frac{4}{3 \sqrt{3}}$ .

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Step 1.1

Multiply $\frac{4}{3 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\cos (x) = \frac{4}{3 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{4}{3 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\cos (x) = \frac{4 \sqrt{3}}{3 \sqrt{3} \sqrt{3}}$

Step 1.2.2

Move $\sqrt{3}$ .

$\cos (x) = \frac{4 \sqrt{3}}{3 (\sqrt{3} \sqrt{3})}$

Step 1.2.3

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

$\cos (x) = \frac{4 \sqrt{3}}{3 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 1.2.4

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

$\cos (x) = \frac{4 \sqrt{3}}{3 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 1.2.5

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

$\cos (x) = \frac{4 \sqrt{3}}{3 {\sqrt{3}}^{1 + 1}}$

Step 1.2.6

Add $1$ and $1$ .

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

Step 1.2.7

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

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Step 1.2.7.1

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

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

Step 1.2.7.2

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

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

Step 1.2.7.3

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

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

Step 1.2.7.4

Cancel the common factor of $2$ .

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Step 1.2.7.4.1

Cancel the common factor.

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

Step 1.2.7.4.2

Rewrite the expression.

$\cos (x) = \frac{4 \sqrt{3}}{3 \cdot 3^{1}}$

Step 1.2.7.5

Evaluate the exponent.

$\cos (x) = \frac{4 \sqrt{3}}{3 \cdot 3}$

Step 1.3

Multiply $3$ by $3$ .

$\cos (x) = \frac{4 \sqrt{3}}{9}$

Step 2

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

$x = \arccos (\frac{4 \sqrt{3}}{9})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arccos (\frac{4 \sqrt{3}}{9})$ .

$x = 0.69226801$

Step 4

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 (3.14159265) - 0.69226801$

Step 5

Simplify $2 (3.14159265) - 0.69226801$ .

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Step 5.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 0.69226801$

Step 5.2

Subtract $0.69226801$ from $6.2831853$ .

$x = 5.59091729$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 1 |}$

Step 6.3

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

$\frac{2 π}{1}$

Step 6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7

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

$x = 0.69226801 + 2 π n, 5.59091729 + 2 π n$ , for any integer $n$