Trigonometry Examples

$- 6 \cos (\frac{π}{2} t) = 2$

Step 1

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

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Step 1.1

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

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

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.

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

Step 1.2.1.2

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

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $2$ and $- 6$ .

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Step 1.3.1.1

Factor $2$ out of $2$ .

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

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $2$ out of $- 6$ .

$\cos (\frac{π}{2} t) = \frac{2 \cdot 1}{2 \cdot - 3}$

Step 1.3.1.2.2

Cancel the common factor.

$\cos (\frac{π}{2} t) = \frac{2 \cdot 1}{2 \cdot - 3}$

Step 1.3.1.2.3

Rewrite the expression.

$\cos (\frac{π}{2} t) = \frac{1}{- 3}$

Step 1.3.2

Move the negative in front of the fraction.

$\cos (\frac{π}{2} t) = - \frac{1}{3}$

Step 2

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

$\frac{π}{2} t = \arccos (- \frac{1}{3})$

Step 3

Simplify the left side.

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Step 3.1

Combine $\frac{π}{2}$ and $t$ .

$\frac{π t}{2} = \arccos (- \frac{1}{3})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arccos (- \frac{1}{3})$ .

$\frac{π t}{2} = 1.91063323$

Step 5

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

$\frac{2}{π} \cdot \frac{π t}{2} = \frac{2}{π} \cdot 1.91063323$

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 $\frac{2}{π} \cdot \frac{π t}{2}$ .

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Step 6.1.1.1

Cancel the common factor of $2$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{2}{π} \cdot \frac{π t}{2} = \frac{2}{π} \cdot 1.91063323$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π t) = \frac{2}{π} \cdot 1.91063323$

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 $π t$ .

$\frac{1}{π} (π (t)) = \frac{2}{π} \cdot 1.91063323$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π t) = \frac{2}{π} \cdot 1.91063323$

Step 6.1.1.2.3

Rewrite the expression.

$t = \frac{2}{π} \cdot 1.91063323$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $\frac{2}{π} \cdot 1.91063323$ .

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Step 6.2.1.1

Multiply $\frac{2}{π} \cdot 1.91063323$ .

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Step 6.2.1.1.1

Combine $\frac{2}{π}$ and $1.91063323$ .

$t = \frac{2 \cdot 1.91063323}{π}$

Step 6.2.1.1.2

Multiply $2$ by $1.91063323$ .

$t = \frac{3.82126647}{π}$

Step 6.2.1.2

Replace $π$ with an approximation.

$t = \frac{3.82126647}{3.14159265}$

Step 6.2.1.3

Divide $3.82126647$ by $3.14159265$ .

$t = 1.21634689$

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.

$\frac{π t}{2} = 2 (3.14159265) - 1.91063323$

Step 8

Solve for $t$ .

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Step 8.1

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

$\frac{2}{π} \cdot \frac{π t}{2} = \frac{2}{π} (2 (3.14159265) - 1.91063323)$

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 $\frac{2}{π} \cdot \frac{π t}{2}$ .

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Step 8.2.1.1.1

Cancel the common factor of $2$ .

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Step 8.2.1.1.1.1

Cancel the common factor.

$\frac{2}{π} \cdot \frac{π t}{2} = \frac{2}{π} (2 (3.14159265) - 1.91063323)$

Step 8.2.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (π t) = \frac{2}{π} (2 (3.14159265) - 1.91063323)$

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 $π t$ .

$\frac{1}{π} (π (t)) = \frac{2}{π} (2 (3.14159265) - 1.91063323)$

Step 8.2.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π t) = \frac{2}{π} (2 (3.14159265) - 1.91063323)$

Step 8.2.1.1.2.3

Rewrite the expression.

$t = \frac{2}{π} (2 (3.14159265) - 1.91063323)$

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

Simplify $\frac{2}{π} (2 (3.14159265) - 1.91063323)$ .

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Step 8.2.2.1.1

Multiply $2$ by $3.14159265$ .

$t = \frac{2}{π} (6.2831853 - 1.91063323)$

Step 8.2.2.1.2

Subtract $1.91063323$ from $6.2831853$ .

$t = \frac{2}{π} \cdot 4.37255207$

Step 8.2.2.1.3

Multiply $\frac{2}{π} \cdot 4.37255207$ .

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Step 8.2.2.1.3.1

Combine $\frac{2}{π}$ and $4.37255207$ .

$t = \frac{2 \cdot 4.37255207}{π}$

Step 8.2.2.1.3.2

Multiply $2$ by $4.37255207$ .

$t = \frac{8.74510414}{π}$

Step 8.2.2.1.4

Replace $π$ with an approximation.

$t = \frac{8.74510414}{3.14159265}$

Step 8.2.2.1.5

Divide $8.74510414$ by $3.14159265$ .

$t = 2.7836531$

Step 9

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

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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 $\frac{π}{2}$ in the formula for period.

$\frac{2 π}{| \frac{π}{2} |}$

Step 9.3

$\frac{π}{2}$ is approximately $1.57079632$ which is positive so remove the absolute value

$\frac{2 π}{\frac{π}{2}}$

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Step 9.5

Cancel the common factor of $π$ .

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Step 9.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Step 9.5.2

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Step 9.5.3

Rewrite the expression.

$2 \cdot 2$

Step 9.6

Multiply $2$ by $2$ .

$4$

Step 10

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

$t = 1.21634689 + 4 n, 2.7836531 + 4 n$ , for any integer $n$