Trigonometry Examples

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

Step 1

Multiply both sides of the equation by $\frac{\sqrt{3}}{2}$ .

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

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{\sqrt{3}}{2} (\frac{2}{\sqrt{3}} \cos (3 x))$ .

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Step 2.1.1.1

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

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

Step 2.1.1.2

Combine and simplify the denominator.

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Step 2.1.1.2.1

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

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

Step 2.1.1.2.2

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

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

Step 2.1.1.2.3

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

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

Step 2.1.1.2.4

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

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

Step 2.1.1.2.5

Add $1$ and $1$ .

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

Step 2.1.1.2.6

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

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Step 2.1.1.2.6.1

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

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

Step 2.1.1.2.6.2

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

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

Step 2.1.1.2.6.3

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

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

Step 2.1.1.2.6.4

Cancel the common factor of $2$ .

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Step 2.1.1.2.6.4.1

Cancel the common factor.

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

Step 2.1.1.2.6.4.2

Rewrite the expression.

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

Step 2.1.1.2.6.5

Evaluate the exponent.

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

Step 2.1.1.3

Combine $\frac{2 \sqrt{3}}{3}$ and $\cos (3 x)$ .

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

Step 2.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.1.4.1

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

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

Step 2.1.1.4.2

Cancel the common factor.

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

Step 2.1.1.4.3

Rewrite the expression.

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

Step 2.1.1.5

Combine $\sqrt{3}$ and $\frac{\sqrt{3} \cos (3 x)}{3}$ .

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

Step 2.1.1.6

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

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

Step 2.1.1.7

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

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

Step 2.1.1.8

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

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

Step 2.1.1.9

Add $1$ and $1$ .

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

Step 2.1.1.10

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

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Step 2.1.1.10.1

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

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

Step 2.1.1.10.2

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

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

Step 2.1.1.10.3

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

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

Step 2.1.1.10.4

Cancel the common factor of $2$ .

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Step 2.1.1.10.4.1

Cancel the common factor.

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

Step 2.1.1.10.4.2

Rewrite the expression.

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

Step 2.1.1.10.5

Evaluate the exponent.

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

Step 2.1.1.11

Cancel the common factor of $3$ .

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Step 2.1.1.11.1

Cancel the common factor.

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

Step 2.1.1.11.2

Divide $\cos (3 x)$ by $1$ .

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $\frac{\sqrt{3}}{2}$ by $1$ .

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

Step 3

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

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

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\arccos (\frac{\sqrt{3}}{2})$ is $\frac{π}{6}$ .

$3 x = \frac{π}{6}$

Step 5

Divide each term in $3 x = \frac{π}{6}$ by $3$ and simplify.

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Step 5.1

Divide each term in $3 x = \frac{π}{6}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{π}{6}}{3}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{π}{6}}{3}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{6}}{3}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{6} \cdot \frac{1}{3}$

Step 5.3.2

Multiply $\frac{π}{6} \cdot \frac{1}{3}$ .

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Step 5.3.2.1

Multiply $\frac{π}{6}$ by $\frac{1}{3}$ .

$x = \frac{π}{6 \cdot 3}$

Step 5.3.2.2

Multiply $6$ by $3$ .

$x = \frac{π}{18}$

Step 6

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.

$3 x = 2 π - \frac{π}{6}$

Step 7

Solve for $x$ .

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Step 7.1

Simplify.

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Step 7.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$3 x = 2 π \cdot \frac{6}{6} - \frac{π}{6}$

Step 7.1.2

Combine $2 π$ and $\frac{6}{6}$ .

$3 x = \frac{2 π \cdot 6}{6} - \frac{π}{6}$

Step 7.1.3

Combine the numerators over the common denominator.

$3 x = \frac{2 π \cdot 6 - π}{6}$

Step 7.1.4

Multiply $6$ by $2$ .

$3 x = \frac{12 π - π}{6}$

Step 7.1.5

Subtract $π$ from $12 π$ .

$3 x = \frac{11 π}{6}$

Step 7.2

Divide each term in $3 x = \frac{11 π}{6}$ by $3$ and simplify.

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Step 7.2.1

Divide each term in $3 x = \frac{11 π}{6}$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{11 π}{6}}{3}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $3$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{11 π}{6}}{3}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{11 π}{6}}{3}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{11 π}{6} \cdot \frac{1}{3}$

Step 7.2.3.2

Multiply $\frac{11 π}{6} \cdot \frac{1}{3}$ .

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Step 7.2.3.2.1

Multiply $\frac{11 π}{6}$ by $\frac{1}{3}$ .

$x = \frac{11 π}{6 \cdot 3}$

Step 7.2.3.2.2

Multiply $6$ by $3$ .

$x = \frac{11 π}{18}$

Step 8

Find the period of $\cos (3 x)$ .

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

Replace $b$ with $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 9

The period of the $\cos (3 x)$ function is $\frac{2 π}{3}$ so values will repeat every $\frac{2 π}{3}$ radians in both directions.

$x = \frac{π}{18} + \frac{2 π n}{3}, \frac{11 π}{18} + \frac{2 π n}{3}$ , for any integer $n$