Trigonometry Examples

2 sin (x) - 1 = 0

$2 \sin (x) - 1 = 0$

Step 1

Add

1

$1$ to both sides of the equation.

2 sin (x) = 1

$2 \sin (x) = 1$

Step 2

Divide each term in

2 sin (x) = 1

$2 \sin (x) = 1$ by

2

$2$ and simplify.

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Step 2.1

Divide each term in

2 sin (x) = 1

$2 \sin (x) = 1$ by

2

$2$ .

\frac{2 sin (x)}{2} = \frac{1}{2}

$\frac{2 \sin (x)}{2} = \frac{1}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of

2

$2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 \sin (x)}{2} = \frac{1}{2}$

Step 2.2.1.2

Divide

$\sin (x)$ by

$1$ .

$\sin (x) = \frac{1}{2}$

Step 3

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

$x$ from inside the sine.

$x = \arcsin (\frac{1}{2})$

Step 4

Simplify the right side.

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Step 4.1

The exact value of

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

$\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 5

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.

$x = π - \frac{π}{6}$

Step 6

Simplify

$π - \frac{π}{6}$ .

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Step 6.1

To write

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

$\frac{6}{6}$ .

$x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 6.2

Combine fractions.

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Step 6.2.1

Combine

$π$ and

$\frac{6}{6}$ .

$x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Move

$6$ to the left of

$π$ .

$x = \frac{6 \cdot π - π}{6}$

Step 6.3.2

Subtract

$π$ from

$6 π$ .

$x = \frac{5 π}{6}$

Step 7

Find the period of

$\sin (x)$ .

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Step 7.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 7.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 7.3

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

$0$ and

$1$ is

$1$ .

$\frac{2 π}{1}$

Step 7.4

Divide

$2 π$ by

$1$ .

$2 π$

Step 8

The period of the

$\sin (x)$ function is

$2 π$ so values will repeat every

$2 π$ radians in both directions.

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer

$n$

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