Trigonometry Examples

$16 = - 10 \cos (\frac{2 π}{150} t) + 10$

Step 1

Rewrite the equation as $- 10 \cos (\frac{2 π}{150} t) + 10 = 16$ .

$- 10 \cos (\frac{2 π}{150} t) + 10 = 16$

Step 2

Move all terms not containing $\cos (\frac{2 π}{150} t)$ to the right side of the equation.

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Step 2.1

Subtract $10$ from both sides of the equation.

$- 10 \cos (\frac{2 π}{150} t) = 16 - 10$

Step 2.2

Subtract $10$ from $16$ .

$- 10 \cos (\frac{2 π}{150} t) = 6$

Step 3

Divide each term in $- 10 \cos (\frac{2 π}{150} t) = 6$ by $- 10$ and simplify.

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Step 3.1

Divide each term in $- 10 \cos (\frac{2 π}{150} t) = 6$ by $- 10$ .

$\frac{- 10 \cos (\frac{2 π}{150} t)}{- 10} = \frac{6}{- 10}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 10$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 10 \cos (\frac{2 π}{150} t)}{- 10} = \frac{6}{- 10}$

Step 3.2.1.2

Divide $\cos (\frac{2 π}{150} t)$ by $1$ .

$\cos (\frac{2 π}{150} t) = \frac{6}{- 10}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $6$ and $- 10$ .

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Step 3.3.1.1

Factor $2$ out of $6$ .

$\cos (\frac{2 π}{150} t) = \frac{2 (3)}{- 10}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $2$ out of $- 10$ .

$\cos (\frac{2 π}{150} t) = \frac{2 \cdot 3}{2 \cdot - 5}$

Step 3.3.1.2.2

Cancel the common factor.

$\cos (\frac{2 π}{150} t) = \frac{2 \cdot 3}{2 \cdot - 5}$

Step 3.3.1.2.3

Rewrite the expression.

$\cos (\frac{2 π}{150} t) = \frac{3}{- 5}$

Step 3.3.2

Move the negative in front of the fraction.

$\cos (\frac{2 π}{150} t) = - \frac{3}{5}$

Step 4

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

$\frac{2 π}{150} t = \arccos (- \frac{3}{5})$

Step 5

Simplify the left side.

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Step 5.1

Simplify $\frac{2 π}{150} t$ .

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Step 5.1.1

Cancel the common factor of $2$ and $150$ .

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Step 5.1.1.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{150} t = \arccos (- \frac{3}{5})$

Step 5.1.1.2

Cancel the common factors.

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Step 5.1.1.2.1

Factor $2$ out of $150$ .

$\frac{2 π}{2 \cdot 75} t = \arccos (- \frac{3}{5})$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 75} t = \arccos (- \frac{3}{5})$

Step 5.1.1.2.3

Rewrite the expression.

$\frac{π}{75} t = \arccos (- \frac{3}{5})$

Step 5.1.2

Combine $\frac{π}{75}$ and $t$ .

$\frac{π t}{75} = \arccos (- \frac{3}{5})$

Step 6

Simplify the right side.

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Step 6.1

Evaluate $\arccos (- \frac{3}{5})$ .

$\frac{π t}{75} = 2.21429743$

Step 7

Multiply both sides of the equation by $\frac{75}{π}$ .

$\frac{75}{π} \cdot \frac{π t}{75} = \frac{75}{π} \cdot 2.21429743$

Step 8

Simplify both sides of the equation.

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Step 8.1

Simplify the left side.

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Step 8.1.1

Simplify $\frac{75}{π} \cdot \frac{π t}{75}$ .

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Step 8.1.1.1

Cancel the common factor of $75$ .

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Step 8.1.1.1.1

Cancel the common factor.

$\frac{75}{π} \cdot \frac{π t}{75} = \frac{75}{π} \cdot 2.21429743$

Step 8.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π t) = \frac{75}{π} \cdot 2.21429743$

Step 8.1.1.2

Cancel the common factor of $π$ .

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Step 8.1.1.2.1

Factor $π$ out of $π t$ .

$\frac{1}{π} (π (t)) = \frac{75}{π} \cdot 2.21429743$

Step 8.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π t) = \frac{75}{π} \cdot 2.21429743$

Step 8.1.1.2.3

Rewrite the expression.

$t = \frac{75}{π} \cdot 2.21429743$

Step 8.2

Simplify the right side.

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Step 8.2.1

Simplify $\frac{75}{π} \cdot 2.21429743$ .

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Step 8.2.1.1

Multiply $\frac{75}{π} \cdot 2.21429743$ .

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Step 8.2.1.1.1

Combine $\frac{75}{π}$ and $2.21429743$ .

$t = \frac{75 \cdot 2.21429743}{π}$

Step 8.2.1.1.2

Multiply $75$ by $2.21429743$ .

$t = \frac{166.07230766}{π}$

Step 8.2.1.2

Replace $π$ with an approximation.

$t = \frac{166.07230766}{3.14159265}$

Step 8.2.1.3

Divide $166.07230766$ by $3.14159265$ .

$t = 52.86245735$

Step 9

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.

$\frac{π t}{75} = 2 (3.14159265) - 2.21429743$

Step 10

Solve for $t$ .

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Step 10.1

Multiply both sides of the equation by $\frac{75}{π}$ .

$\frac{75}{π} \cdot \frac{π t}{75} = \frac{75}{π} (2 (3.14159265) - 2.21429743)$

Step 10.2

Simplify both sides of the equation.

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Step 10.2.1

Simplify the left side.

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Step 10.2.1.1

Simplify $\frac{75}{π} \cdot \frac{π t}{75}$ .

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Step 10.2.1.1.1

Cancel the common factor of $75$ .

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Step 10.2.1.1.1.1

Cancel the common factor.

$\frac{75}{π} \cdot \frac{π t}{75} = \frac{75}{π} (2 (3.14159265) - 2.21429743)$

Step 10.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π t) = \frac{75}{π} (2 (3.14159265) - 2.21429743)$

Step 10.2.1.1.2

Cancel the common factor of $π$ .

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Step 10.2.1.1.2.1

Factor $π$ out of $π t$ .

$\frac{1}{π} (π (t)) = \frac{75}{π} (2 (3.14159265) - 2.21429743)$

Step 10.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π t) = \frac{75}{π} (2 (3.14159265) - 2.21429743)$

Step 10.2.1.1.2.3

Rewrite the expression.

$t = \frac{75}{π} (2 (3.14159265) - 2.21429743)$

Step 10.2.2

Simplify the right side.

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Step 10.2.2.1

Simplify $\frac{75}{π} (2 (3.14159265) - 2.21429743)$ .

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Step 10.2.2.1.1

Multiply $2$ by $3.14159265$ .

$t = \frac{75}{π} (6.2831853 - 2.21429743)$

Step 10.2.2.1.2

Subtract $2.21429743$ from $6.2831853$ .

$t = \frac{75}{π} \cdot 4.06888787$

Step 10.2.2.1.3

Multiply $\frac{75}{π} \cdot 4.06888787$ .

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Step 10.2.2.1.3.1

Combine $\frac{75}{π}$ and $4.06888787$ .

$t = \frac{75 \cdot 4.06888787}{π}$

Step 10.2.2.1.3.2

Multiply $75$ by $4.06888787$ .

$t = \frac{305.16659036}{π}$

Step 10.2.2.1.4

Replace $π$ with an approximation.

$t = \frac{305.16659036}{3.14159265}$

Step 10.2.2.1.5

Divide $305.16659036$ by $3.14159265$ .

$t = 97.13754264$

Step 11

Find the period of $\cos (\frac{2 π}{150} t)$ .

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Step 11.1

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

$\frac{2 π}{| b |}$

Step 11.2

Replace $b$ with $\frac{π}{75}$ in the formula for period.

$\frac{2 π}{| \frac{π}{75} |}$

Step 11.3

$\frac{π}{75}$ is approximately $0.0418879$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{75}}$

Step 11.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{75}{π}$

Step 11.5

Cancel the common factor of $π$ .

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Step 11.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{75}{π}$

Step 11.5.2

Cancel the common factor.

$π \cdot 2 \frac{75}{π}$

Step 11.5.3

Rewrite the expression.

$2 \cdot 75$

Step 11.6

Multiply $2$ by $75$ .

$150$

Step 12

The period of the $\cos (\frac{2 π}{150} t)$ function is $150$ so values will repeat every $150$ radians in both directions.

$t = 52.86245735 + 150 n, 97.13754264 + 150 n$ , for any integer $n$