Trigonometry Examples

sin (3 t) = \frac{1}{2}

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

Step 1

Take the inverse sine of both sides of the equation to extract

t

$t$ from inside the sine.

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

Step 2

Simplify the right side.

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Step 2.1

The exact value of

$\arcsin (\frac{1}{2})$ is

$\frac{π}{6}$ .

$3 t = \frac{π}{6}$

Step 3

Divide each term in

$3 t = \frac{π}{6}$ by

$3$ and simplify.

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Step 3.1

Divide each term in

$3 t = \frac{π}{6}$ by

$3$ .

$\frac{3 t}{3} = \frac{\frac{π}{6}}{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

$3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 t}{3} = \frac{\frac{π}{6}}{3}$

Step 3.2.1.2

Divide

$t$ by

$1$ .

$t = \frac{\frac{π}{6}}{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$t = \frac{π}{6} \cdot \frac{1}{3}$

Step 3.3.2

Multiply

$\frac{π}{6} \cdot \frac{1}{3}$ .

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Step 3.3.2.1

Multiply

$\frac{π}{6}$ by

$\frac{1}{3}$ .

$t = \frac{π}{6 \cdot 3}$

Step 3.3.2.2

Multiply

$6$ by

$3$ .

$t = \frac{π}{18}$

Step 4

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from

$π$ to find the solution in the second quadrant.

$3 t = π - \frac{π}{6}$

Step 5

Solve for

$t$ .

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Step 5.1

Simplify.

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Step 5.1.1

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{6}{6}$ .

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

Step 5.1.2

Combine

$π$ and

$\frac{6}{6}$ .

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

Step 5.1.3

Combine the numerators over the common denominator.

$3 t = \frac{π \cdot 6 - π}{6}$

Step 5.1.4

Subtract

$π$ from

$π \cdot 6$ .

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Step 5.1.4.1

Reorder

$π$ and

$6$ .

$3 t = \frac{6 \cdot π - π}{6}$

Step 5.1.4.2

Subtract

$π$ from

$6 \cdot π$ .

$3 t = \frac{5 \cdot π}{6}$

Step 5.2

Divide each term in

$3 t = \frac{5 \cdot π}{6}$ by

$3$ and simplify.

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Step 5.2.1

Divide each term in

$3 t = \frac{5 \cdot π}{6}$ by

$3$ .

$\frac{3 t}{3} = \frac{\frac{5 \cdot π}{6}}{3}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of

$3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{3 t}{3} = \frac{\frac{5 \cdot π}{6}}{3}$

Step 5.2.2.1.2

Divide

$t$ by

$1$ .

$t = \frac{\frac{5 \cdot π}{6}}{3}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$t = \frac{5 \cdot π}{6} \cdot \frac{1}{3}$

Step 5.2.3.2

Multiply

$\frac{5 π}{6} \cdot \frac{1}{3}$ .

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Step 5.2.3.2.1

Multiply

$\frac{5 π}{6}$ by

$\frac{1}{3}$ .

$t = \frac{5 π}{6 \cdot 3}$

Step 5.2.3.2.2

Multiply

$6$ by

$3$ .

$t = \frac{5 π}{18}$

Step 6

Find the period of

$\sin (3 t)$ .

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Step 6.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 6.2

Replace

$b$ with

$3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 6.3

The absolute value is the distance between a number and zero. The distance between

$0$ and

$3$ is

$3$ .

$\frac{2 π}{3}$

Step 7

The period of the

$\sin (3 t)$ function is

$\frac{2 π}{3}$ so values will repeat every

$\frac{2 π}{3}$ radians in both directions.

$t = \frac{π}{18} + \frac{2 π n}{3}, \frac{5 π}{18} + \frac{2 π n}{3}$ , for any integer

$n$