Precalculus Examples

sin (θ) = 0

$\sin (θ) = 0$

Step 1

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

θ

$θ$ from inside the sine.

θ = arcsin (0)

$θ = \arcsin (0)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of

arcsin (0)

$\arcsin (0)$ is

0

$0$ .

θ = 0

$θ = 0$

θ = 0

$θ = 0$

Step 3

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

π

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

θ = π - 0

$θ = π - 0$

Step 4

Subtract

0

$0$ from

π

$π$ .

θ = π

$θ = π$

Step 5

Find the period of

sin (θ)

$\sin (θ)$ .

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Step 5.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 5.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 5.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 5.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 6

The period of the

sin (θ)

$\sin (θ)$ function is

2 π

$2 π$ so values will repeat every

2 π

$2 π$ radians in both directions.

θ = 2 π n, π + 2 π n

$θ = 2 π n, π + 2 π n$ , for any integer

n

$n$

Step 7

Consolidate the answers.

θ = π n

$θ = π n$ , for any integer

n

$n$