Trigonometry Examples

$- 6 \cos (\frac{π}{3} x) = 4$

Step 1

Divide each term in $- 6 \cos (\frac{π}{3} x) = 4$ by $- 6$ and simplify.

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Step 1.1

Divide each term in $- 6 \cos (\frac{π}{3} x) = 4$ by $- 6$ .

$\frac{- 6 \cos (\frac{π}{3} x)}{- 6} = \frac{4}{- 6}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $- 6$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{- 6 \cos (\frac{π}{3} x)}{- 6} = \frac{4}{- 6}$

Step 1.2.1.2

Divide $\cos (\frac{π}{3} x)$ by $1$ .

$\cos (\frac{π}{3} x) = \frac{4}{- 6}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $4$ and $- 6$ .

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Step 1.3.1.1

Factor $2$ out of $4$ .

$\cos (\frac{π}{3} x) = \frac{2 (2)}{- 6}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $2$ out of $- 6$ .

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

Step 1.3.1.2.2

Cancel the common factor.

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

Step 1.3.1.2.3

Rewrite the expression.

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

Step 1.3.2

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

Simplify the left side.

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Step 3.1

Combine $\frac{π}{3}$ and $x$ .

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

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arccos (- \frac{2}{3})$ .

$\frac{π x}{3} = 2.30052398$

Step 5

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

$\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} \cdot 2.30052398$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

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

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Step 6.1.1.1

Cancel the common factor of $3$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} \cdot 2.30052398$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{3}{π} \cdot 2.30052398$

Step 6.1.1.2

Cancel the common factor of $π$ .

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Step 6.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{3}{π} \cdot 2.30052398$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{3}{π} \cdot 2.30052398$

Step 6.1.1.2.3

Rewrite the expression.

$x = \frac{3}{π} \cdot 2.30052398$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{3}{π} \cdot 2.30052398$ .

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Step 6.2.1.1

Multiply $\frac{3}{π} \cdot 2.30052398$ .

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Step 6.2.1.1.1

Combine $\frac{3}{π}$ and $2.30052398$ .

$x = \frac{3 \cdot 2.30052398}{π}$

Step 6.2.1.1.2

Multiply $3$ by $2.30052398$ .

$x = \frac{6.90157194}{π}$

Step 6.2.1.2

Replace $π$ with an approximation.

$x = \frac{6.90157194}{3.14159265}$

Step 6.2.1.3

Divide $6.90157194$ by $3.14159265$ .

$x = 2.19683858$

Step 7

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

$\frac{π x}{3} = 2 (3.14159265) - 2.30052398$

Step 8

Solve for $x$ .

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Step 8.1

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

$\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} (2 (3.14159265) - 2.30052398)$

Step 8.2

Simplify both sides of the equation.

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Step 8.2.1

Simplify the left side.

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Step 8.2.1.1

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

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Step 8.2.1.1.1

Cancel the common factor of $3$ .

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Step 8.2.1.1.1.1

Cancel the common factor.

$\frac{3}{π} \cdot \frac{π x}{3} = \frac{3}{π} (2 (3.14159265) - 2.30052398)$

Step 8.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π x) = \frac{3}{π} (2 (3.14159265) - 2.30052398)$

Step 8.2.1.1.2

Cancel the common factor of $π$ .

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Step 8.2.1.1.2.1

Factor $π$ out of $π x$ .

$\frac{1}{π} (π (x)) = \frac{3}{π} (2 (3.14159265) - 2.30052398)$

Step 8.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{3}{π} (2 (3.14159265) - 2.30052398)$

Step 8.2.1.1.2.3

Rewrite the expression.

$x = \frac{3}{π} (2 (3.14159265) - 2.30052398)$

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

Simplify $\frac{3}{π} (2 (3.14159265) - 2.30052398)$ .

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Step 8.2.2.1.1

Multiply $2$ by $3.14159265$ .

$x = \frac{3}{π} (6.2831853 - 2.30052398)$

Step 8.2.2.1.2

Subtract $2.30052398$ from $6.2831853$ .

$x = \frac{3}{π} \cdot 3.98266132$

Step 8.2.2.1.3

Multiply $\frac{3}{π} \cdot 3.98266132$ .

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Step 8.2.2.1.3.1

Combine $\frac{3}{π}$ and $3.98266132$ .

$x = \frac{3 \cdot 3.98266132}{π}$

Step 8.2.2.1.3.2

Multiply $3$ by $3.98266132$ .

$x = \frac{11.94798397}{π}$

Step 8.2.2.1.4

Replace $π$ with an approximation.

$x = \frac{11.94798397}{3.14159265}$

Step 8.2.2.1.5

Divide $11.94798397$ by $3.14159265$ .

$x = 3.80316141$

Step 9

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

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Step 9.1

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

$\frac{2 π}{| b |}$

Step 9.2

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

$\frac{2 π}{| \frac{π}{3} |}$

Step 9.3

$\frac{π}{3}$ is approximately $1.04719755$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{3}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{π}$

Step 9.5

Cancel the common factor of $π$ .

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Step 9.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{3}{π}$

Step 9.5.2

Cancel the common factor.

$π \cdot 2 \frac{3}{π}$

Step 9.5.3

Rewrite the expression.

$2 \cdot 3$

Step 9.6

Multiply $2$ by $3$ .

$6$

Step 10

The period of the $\cos (\frac{π}{3} x)$ function is $6$ so values will repeat every $6$ radians in both directions.

$x = 2.19683858 + 6 n, 3.80316141 + 6 n$ , for any integer $n$