Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify the left side.

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Step 2.1

Combine $\frac{π}{2}$ and $t$ .

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

Step 3

Simplify the right side.

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Step 3.1

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

$\frac{π t}{2} = \frac{2 π}{3}$

Step 4

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

$\frac{2}{π} \cdot \frac{π t}{2} = \frac{2}{π} \cdot \frac{2 π}{3}$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

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

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Step 5.1.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1.1

Cancel the common factor.

$\frac{2}{π} \cdot \frac{π t}{2} = \frac{2}{π} \cdot \frac{2 π}{3}$

Step 5.1.1.1.2

Rewrite the expression.

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

Step 5.1.1.2

Cancel the common factor of $π$ .

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Step 5.1.1.2.1

Factor $π$ out of $π t$ .

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

Step 5.1.1.2.2

Cancel the common factor.

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

Step 5.1.1.2.3

Rewrite the expression.

$t = \frac{2}{π} \cdot \frac{2 π}{3}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{2}{π} \cdot \frac{2 π}{3}$ .

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Step 5.2.1.1

Cancel the common factor of $π$ .

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Step 5.2.1.1.1

Factor $π$ out of $2 π$ .

$t = \frac{2}{π} \cdot \frac{π \cdot 2}{3}$

Step 5.2.1.1.2

Cancel the common factor.

$t = \frac{2}{π} \cdot \frac{π \cdot 2}{3}$

Step 5.2.1.1.3

Rewrite the expression.

$t = 2 (\frac{2}{3})$

Step 5.2.1.2

Combine $2$ and $\frac{2}{3}$ .

$t = \frac{2 \cdot 2}{3}$

Step 5.2.1.3

Multiply $2$ by $2$ .

$t = \frac{4}{3}$

Step 6

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.

$\frac{π t}{2} = 2 π - \frac{2 π}{3}$

Step 7

Solve for $t$ .

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Step 7.1

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

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

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

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

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Step 7.2.1.1.1

Cancel the common factor of $2$ .

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Step 7.2.1.1.1.1

Cancel the common factor.

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

Step 7.2.1.1.1.2

Rewrite the expression.

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

Step 7.2.1.1.2

Cancel the common factor of $π$ .

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Step 7.2.1.1.2.1

Factor $π$ out of $π t$ .

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

Step 7.2.1.1.2.2

Cancel the common factor.

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

Step 7.2.1.1.2.3

Rewrite the expression.

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

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

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

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Step 7.2.2.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 7.2.2.1.2

Combine fractions.

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Step 7.2.2.1.2.1

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 7.2.2.1.2.2

Combine the numerators over the common denominator.

$t = \frac{2}{π} \cdot \frac{2 π \cdot 3 - 2 π}{3}$

Step 7.2.2.1.3

Simplify the numerator.

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Step 7.2.2.1.3.1

Multiply $3$ by $2$ .

$t = \frac{2}{π} \cdot \frac{6 π - 2 π}{3}$

Step 7.2.2.1.3.2

Subtract $2 π$ from $6 π$ .

$t = \frac{2}{π} \cdot \frac{4 π}{3}$

Step 7.2.2.1.4

Simplify terms.

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Step 7.2.2.1.4.1

Cancel the common factor of $π$ .

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Step 7.2.2.1.4.1.1

Factor $π$ out of $4 π$ .

$t = \frac{2}{π} \cdot \frac{π \cdot 4}{3}$

Step 7.2.2.1.4.1.2

Cancel the common factor.

$t = \frac{2}{π} \cdot \frac{π \cdot 4}{3}$

Step 7.2.2.1.4.1.3

Rewrite the expression.

$t = 2 (\frac{4}{3})$

Step 7.2.2.1.4.2

Combine $2$ and $\frac{4}{3}$ .

$t = \frac{2 \cdot 4}{3}$

Step 7.2.2.1.4.3

Multiply $2$ by $4$ .

$t = \frac{8}{3}$

Step 8

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

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

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

$\frac{2 π}{| \frac{π}{2} |}$

Step 8.3

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

$\frac{2 π}{\frac{π}{2}}$

Step 8.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{π}$

Step 8.5

Cancel the common factor of $π$ .

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Step 8.5.1

Factor $π$ out of $2 π$ .

$π \cdot 2 \frac{2}{π}$

Step 8.5.2

Cancel the common factor.

$π \cdot 2 \frac{2}{π}$

Step 8.5.3

Rewrite the expression.

$2 \cdot 2$

Step 8.6

Multiply $2$ by $2$ .

$4$

Step 9

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

$t = \frac{4}{3} + 4 n, \frac{8}{3} + 4 n$ , for any integer $n$