Trigonometry Examples

$\cos (\frac{4 x}{3}) = 0.3$

Step 1

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

$\frac{4 x}{3} = \arccos (0.3)$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arccos (0.3)$ .

$\frac{4 x}{3} = 1.26610367$

Step 3

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} \cdot \frac{4 x}{3} = \frac{3}{4} \cdot 1.26610367$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{3}{4} \cdot \frac{4 x}{3}$ .

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Step 4.1.1.1

Cancel the common factor of $3$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{3}{4} \cdot \frac{4 x}{3} = \frac{3}{4} \cdot 1.26610367$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{4} (4 x) = \frac{3}{4} \cdot 1.26610367$

Step 4.1.1.2

Cancel the common factor of $4$ .

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Step 4.1.1.2.1

Factor $4$ out of $4 x$ .

$\frac{1}{4} (4 (x)) = \frac{3}{4} \cdot 1.26610367$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{4} (4 x) = \frac{3}{4} \cdot 1.26610367$

Step 4.1.1.2.3

Rewrite the expression.

$x = \frac{3}{4} \cdot 1.26610367$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3}{4} \cdot 1.26610367$ .

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Step 4.2.1.1

Multiply $\frac{3}{4} \cdot 1.26610367$ .

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Step 4.2.1.1.1

Combine $\frac{3}{4}$ and $1.26610367$ .

$x = \frac{3 \cdot 1.26610367}{4}$

Step 4.2.1.1.2

Multiply $3$ by $1.26610367$ .

$x = \frac{3.79831101}{4}$

Step 4.2.1.2

Divide $3.79831101$ by $4$ .

$x = 0.94957775$

Step 5

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.

$\frac{4 x}{3} = 2 (3.14159265) - 1.26610367$

Step 6

Solve for $x$ .

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Step 6.1

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} \cdot \frac{4 x}{3} = \frac{3}{4} (2 (3.14159265) - 1.26610367)$

Step 6.2

Simplify both sides of the equation.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify $\frac{3}{4} \cdot \frac{4 x}{3}$ .

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Step 6.2.1.1.1

Cancel the common factor of $3$ .

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Step 6.2.1.1.1.1

Cancel the common factor.

$\frac{3}{4} \cdot \frac{4 x}{3} = \frac{3}{4} (2 (3.14159265) - 1.26610367)$

Step 6.2.1.1.1.2

Rewrite the expression.

$\frac{1}{4} (4 x) = \frac{3}{4} (2 (3.14159265) - 1.26610367)$

Step 6.2.1.1.2

Cancel the common factor of $4$ .

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Step 6.2.1.1.2.1

Factor $4$ out of $4 x$ .

$\frac{1}{4} (4 (x)) = \frac{3}{4} (2 (3.14159265) - 1.26610367)$

Step 6.2.1.1.2.2

Cancel the common factor.

$\frac{1}{4} (4 x) = \frac{3}{4} (2 (3.14159265) - 1.26610367)$

Step 6.2.1.1.2.3

Rewrite the expression.

$x = \frac{3}{4} (2 (3.14159265) - 1.26610367)$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $\frac{3}{4} (2 (3.14159265) - 1.26610367)$ .

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Step 6.2.2.1.1

Multiply $2$ by $3.14159265$ .

$x = \frac{3}{4} (6.2831853 - 1.26610367)$

Step 6.2.2.1.2

Subtract $1.26610367$ from $6.2831853$ .

$x = \frac{3}{4} \cdot 5.01708163$

Step 6.2.2.1.3

Multiply $\frac{3}{4} \cdot 5.01708163$ .

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Step 6.2.2.1.3.1

Combine $\frac{3}{4}$ and $5.01708163$ .

$x = \frac{3 \cdot 5.01708163}{4}$

Step 6.2.2.1.3.2

Multiply $3$ by $5.01708163$ .

$x = \frac{15.0512449}{4}$

Step 6.2.2.1.4

Divide $15.0512449$ by $4$ .

$x = 3.76281122$

Step 7

Find the period of $\cos (\frac{4 x}{3})$ .

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

Replace $b$ with $\frac{4}{3}$ in the formula for period.

$\frac{2 π}{| \frac{4}{3} |}$

Step 7.3

$\frac{4}{3}$ is approximately $1.\overline{3}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{4}{3}}$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{4}$

Step 7.5

Cancel the common factor of $2$ .

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Step 7.5.1

Factor $2$ out of $2 π$ .

$2 (π) \frac{3}{4}$

Step 7.5.2

Factor $2$ out of $4$ .

$2 (π) \frac{3}{2 (2)}$

Step 7.5.3

Cancel the common factor.

$2 π \frac{3}{2 \cdot 2}$

Step 7.5.4

Rewrite the expression.

$π \frac{3}{2}$

Step 7.6

Combine $π$ and $\frac{3}{2}$ .

$\frac{π \cdot 3}{2}$

Step 7.7

Move $3$ to the left of $π$ .

$\frac{3 π}{2}$

Step 8

The period of the $\cos (\frac{4 x}{3})$ function is $\frac{3 π}{2}$ so values will repeat every $\frac{3 π}{2}$ radians in both directions.

$x = 0.94957775 + \frac{3 π n}{2}, 3.76281122 + \frac{3 π n}{2}$ , for any integer $n$