Trigonometry Examples

$6 = - 8 \cos (\frac{π}{6} t)$

Step 1

Rewrite the equation as $- 8 \cos (\frac{π}{6} t) = 6$ .

$- 8 \cos (\frac{π}{6} t) = 6$

Step 2

Divide each term in $- 8 \cos (\frac{π}{6} t) = 6$ by $- 8$ and simplify.

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Step 2.1

Divide each term in $- 8 \cos (\frac{π}{6} t) = 6$ by $- 8$ .

$\frac{- 8 \cos (\frac{π}{6} t)}{- 8} = \frac{6}{- 8}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 8$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 8 \cos (\frac{π}{6} t)}{- 8} = \frac{6}{- 8}$

Step 2.2.1.2

Divide $\cos (\frac{π}{6} t)$ by $1$ .

$\cos (\frac{π}{6} t) = \frac{6}{- 8}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $6$ and $- 8$ .

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Step 2.3.1.1

Factor $2$ out of $6$ .

$\cos (\frac{π}{6} t) = \frac{2 (3)}{- 8}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $2$ out of $- 8$ .

$\cos (\frac{π}{6} t) = \frac{2 \cdot 3}{2 \cdot - 4}$

Step 2.3.1.2.2

Cancel the common factor.

$\cos (\frac{π}{6} t) = \frac{2 \cdot 3}{2 \cdot - 4}$

Step 2.3.1.2.3

Rewrite the expression.

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

Step 2.3.2

Move the negative in front of the fraction.

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

Step 3

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

$\frac{π}{6} t = \arccos (- \frac{3}{4})$

Step 4

Simplify the left side.

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Step 4.1

Combine $\frac{π}{6}$ and $t$ .

$\frac{π t}{6} = \arccos (- \frac{3}{4})$

Step 5

Simplify the right side.

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Step 5.1

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

$\frac{π t}{6} = 2.4188584$

Step 6

Multiply both sides of the equation by $\frac{6}{π}$ .

$\frac{6}{π} \cdot \frac{π t}{6} = \frac{6}{π} \cdot 2.4188584$

Step 7

Simplify both sides of the equation.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Simplify $\frac{6}{π} \cdot \frac{π t}{6}$ .

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Step 7.1.1.1

Cancel the common factor of $6$ .

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Step 7.1.1.1.1

Cancel the common factor.

$\frac{6}{π} \cdot \frac{π t}{6} = \frac{6}{π} \cdot 2.4188584$

Step 7.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π t) = \frac{6}{π} \cdot 2.4188584$

Step 7.1.1.2

Cancel the common factor of $π$ .

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Step 7.1.1.2.1

Factor $π$ out of $π t$ .

$\frac{1}{π} (π (t)) = \frac{6}{π} \cdot 2.4188584$

Step 7.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π t) = \frac{6}{π} \cdot 2.4188584$

Step 7.1.1.2.3

Rewrite the expression.

$t = \frac{6}{π} \cdot 2.4188584$

Step 7.2

Simplify the right side.

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Step 7.2.1

Simplify $\frac{6}{π} \cdot 2.4188584$ .

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Step 7.2.1.1

Multiply $\frac{6}{π} \cdot 2.4188584$ .

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Step 7.2.1.1.1

Combine $\frac{6}{π}$ and $2.4188584$ .

$t = \frac{6 \cdot 2.4188584}{π}$

Step 7.2.1.1.2

Multiply $6$ by $2.4188584$ .

$t = \frac{14.51315043}{π}$

Step 7.2.1.2

Replace $π$ with an approximation.

$t = \frac{14.51315043}{3.14159265}$

Step 7.2.1.3

Divide $14.51315043$ by $3.14159265$ .

$t = 4.61967926$

Step 8

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.

$\frac{π t}{6} = 2 (3.14159265) - 2.4188584$

Step 9

Solve for $t$ .

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Step 9.1

Multiply both sides of the equation by $\frac{6}{π}$ .

$\frac{6}{π} \cdot \frac{π t}{6} = \frac{6}{π} (2 (3.14159265) - 2.4188584)$

Step 9.2

Simplify both sides of the equation.

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Step 9.2.1

Simplify the left side.

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Step 9.2.1.1

Simplify $\frac{6}{π} \cdot \frac{π t}{6}$ .

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Step 9.2.1.1.1

Cancel the common factor of $6$ .

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Step 9.2.1.1.1.1

Cancel the common factor.

$\frac{6}{π} \cdot \frac{π t}{6} = \frac{6}{π} (2 (3.14159265) - 2.4188584)$

Step 9.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π t) = \frac{6}{π} (2 (3.14159265) - 2.4188584)$

Step 9.2.1.1.2

Cancel the common factor of $π$ .

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Step 9.2.1.1.2.1

Factor $π$ out of $π t$ .

$\frac{1}{π} (π (t)) = \frac{6}{π} (2 (3.14159265) - 2.4188584)$

Step 9.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π t) = \frac{6}{π} (2 (3.14159265) - 2.4188584)$

Step 9.2.1.1.2.3

Rewrite the expression.

$t = \frac{6}{π} (2 (3.14159265) - 2.4188584)$

Step 9.2.2

Simplify the right side.

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Step 9.2.2.1

Simplify $\frac{6}{π} (2 (3.14159265) - 2.4188584)$ .

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Step 9.2.2.1.1

Multiply $2$ by $3.14159265$ .

$t = \frac{6}{π} (6.2831853 - 2.4188584)$

Step 9.2.2.1.2

Subtract $2.4188584$ from $6.2831853$ .

$t = \frac{6}{π} \cdot 3.8643269$

Step 9.2.2.1.3

Multiply $\frac{6}{π} \cdot 3.8643269$ .

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Step 9.2.2.1.3.1

Combine $\frac{6}{π}$ and $3.8643269$ .

$t = \frac{6 \cdot 3.8643269}{π}$

Step 9.2.2.1.3.2

Multiply $6$ by $3.8643269$ .

$t = \frac{23.1859614}{π}$

Step 9.2.2.1.4

Replace $π$ with an approximation.

$t = \frac{23.1859614}{3.14159265}$

Step 9.2.2.1.5

Divide $23.1859614$ by $3.14159265$ .

$t = 7.38032073$

Step 10

Find the period of $\cos (\frac{π}{6} t)$ .

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Step 10.1

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

$\frac{2 π}{| b |}$

Step 10.2

Replace $b$ with $\frac{π}{6}$ in the formula for period.

$\frac{2 π}{| \frac{π}{6} |}$

Step 10.3

$\frac{π}{6}$ is approximately $0.52359877$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{6}}$

Step 10.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{6}{π}$

Step 10.5

Cancel the common factor of $π$ .

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Step 10.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{6}{π}$

Step 10.5.2

Cancel the common factor.

$π \cdot 2 \frac{6}{π}$

Step 10.5.3

Rewrite the expression.

$2 \cdot 6$

Step 10.6

Multiply $2$ by $6$ .

$12$

Step 11

The period of the $\cos (\frac{π}{6} t)$ function is $12$ so values will repeat every $12$ radians in both directions.

$t = 4.61967926 + 12 n, 7.38032073 + 12 n$ , for any integer $n$