Trigonometry Examples

2 \sin (2 x) - 1 = 0

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

[0, 2 π)

$[0, 2 π)$

Step 1

Add

1

$1$ to both sides of the equation.

2 \sin (2 x) = 1

$2 \sin (2 x) = 1$

Step 2

Divide each term in

2 \sin (2 x) = 1

$2 \sin (2 x) = 1$ by

2

$2$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in

2 \sin (2 x) = 1

$2 \sin (2 x) = 1$ by

2

$2$ .

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

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

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

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

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

Step 2.2.1.2

Divide

\sin (2 x)

$\sin (2 x)$ by

1

$1$ .

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

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

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

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

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

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

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

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

Step 3

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

x

$x$ from inside the sine.

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

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

Step 4

Simplify the right side.

Tap for more steps...

Step 4.1

The exact value of

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

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

\frac{π}{6}

$\frac{π}{6}$ .

2 x = \frac{π}{6}

$2 x = \frac{π}{6}$

2 x = \frac{π}{6}

$2 x = \frac{π}{6}$

Step 5

Divide each term in

2 x = \frac{π}{6}

$2 x = \frac{π}{6}$ by

2

$2$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in

2 x = \frac{π}{6}

$2 x = \frac{π}{6}$ by

2

$2$ .

\frac{2 x}{2} = \frac{\frac{π}{6}}{2}

$\frac{2 x}{2} = \frac{\frac{π}{6}}{2}$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

\frac{2 x}{2} = \frac{\frac{π}{6}}{2}

$\frac{2 x}{2} = \frac{\frac{π}{6}}{2}$

Step 5.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{\frac{π}{6}}{2}

$x = \frac{\frac{π}{6}}{2}$

x = \frac{\frac{π}{6}}{2}

$x = \frac{\frac{π}{6}}{2}$

x = \frac{\frac{π}{6}}{2}

$x = \frac{\frac{π}{6}}{2}$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

x = \frac{π}{6} \cdot \frac{1}{2}

$x = \frac{π}{6} \cdot \frac{1}{2}$

Step 5.3.2

Multiply

\frac{π}{6} \cdot \frac{1}{2}

$\frac{π}{6} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 5.3.2.1

Multiply

\frac{π}{6}

$\frac{π}{6}$ by

\frac{1}{2}

$\frac{1}{2}$ .

x = \frac{π}{6 \cdot 2}

$x = \frac{π}{6 \cdot 2}$

Step 5.3.2.2

Multiply

6

$6$ by

2

$2$ .

x = \frac{π}{12}

$x = \frac{π}{12}$

x = \frac{π}{12}

$x = \frac{π}{12}$

x = \frac{π}{12}

$x = \frac{π}{12}$

x = \frac{π}{12}

$x = \frac{π}{12}$

Step 6

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.

2 x = π - \frac{π}{6}

$2 x = π - \frac{π}{6}$

Step 7

Solve for

x

$x$ .

Tap for more steps...

Step 7.1

Simplify.

Tap for more steps...

Step 7.1.1

To write

π

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

\frac{6}{6}

$\frac{6}{6}$ .

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

$2 x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 7.1.2

Combine

π

$π$ and

\frac{6}{6}

$\frac{6}{6}$ .

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

$2 x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 7.1.3

Combine the numerators over the common denominator.

2 x = \frac{π \cdot 6 - π}{6}

$2 x = \frac{π \cdot 6 - π}{6}$

Step 7.1.4

Subtract

π

$π$ from

π \cdot 6

$π \cdot 6$ .

Tap for more steps...

Step 7.1.4.1

Reorder

π

$π$ and

6

$6$ .

2 x = \frac{6 \cdot π - π}{6}

$2 x = \frac{6 \cdot π - π}{6}$

Step 7.1.4.2

Subtract

π

$π$ from

6 \cdot π

$6 \cdot π$ .

2 x = \frac{5 \cdot π}{6}

$2 x = \frac{5 \cdot π}{6}$

2 x = \frac{5 \cdot π}{6}

$2 x = \frac{5 \cdot π}{6}$

2 x = \frac{5 \cdot π}{6}

$2 x = \frac{5 \cdot π}{6}$

Step 7.2

Divide each term in

2 x = \frac{5 \cdot π}{6}

$2 x = \frac{5 \cdot π}{6}$ by

2

$2$ and simplify.

Tap for more steps...

Step 7.2.1

Divide each term in

2 x = \frac{5 \cdot π}{6}

$2 x = \frac{5 \cdot π}{6}$ by

2

$2$ .

\frac{2 x}{2} = \frac{\frac{5 \cdot π}{6}}{2}

$\frac{2 x}{2} = \frac{\frac{5 \cdot π}{6}}{2}$

Step 7.2.2

Simplify the left side.

Tap for more steps...

Step 7.2.2.1

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 7.2.2.1.1

Cancel the common factor.

\frac{2 x}{2} = \frac{\frac{5 \cdot π}{6}}{2}

$\frac{2 x}{2} = \frac{\frac{5 \cdot π}{6}}{2}$

Step 7.2.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{\frac{5 \cdot π}{6}}{2}

$x = \frac{\frac{5 \cdot π}{6}}{2}$

x = \frac{\frac{5 \cdot π}{6}}{2}

$x = \frac{\frac{5 \cdot π}{6}}{2}$

x = \frac{\frac{5 \cdot π}{6}}{2}

$x = \frac{\frac{5 \cdot π}{6}}{2}$

Step 7.2.3

Simplify the right side.

Tap for more steps...

Step 7.2.3.1

Multiply the numerator by the reciprocal of the denominator.

x = \frac{5 \cdot π}{6} \cdot \frac{1}{2}

$x = \frac{5 \cdot π}{6} \cdot \frac{1}{2}$

Step 7.2.3.2

Multiply

\frac{5 π}{6} \cdot \frac{1}{2}

$\frac{5 π}{6} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 7.2.3.2.1

Multiply

\frac{5 π}{6}

$\frac{5 π}{6}$ by

\frac{1}{2}

$\frac{1}{2}$ .

x = \frac{5 π}{6 \cdot 2}

$x = \frac{5 π}{6 \cdot 2}$

Step 7.2.3.2.2

Multiply

6

$6$ by

2

$2$ .

x = \frac{5 π}{12}

$x = \frac{5 π}{12}$

x = \frac{5 π}{12}

$x = \frac{5 π}{12}$

x = \frac{5 π}{12}

$x = \frac{5 π}{12}$

x = \frac{5 π}{12}

$x = \frac{5 π}{12}$

x = \frac{5 π}{12}

$x = \frac{5 π}{12}$

Step 8

Find the period of

\sin (2 x)

$\sin (2 x)$ .

Tap for more steps...

Step 8.1

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 8.2

Replace

b

$b$ with

2

$2$ in the formula for period.

\frac{2 π}{| 2 |}

$\frac{2 π}{| 2 |}$

Step 8.3

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

0

$0$ and

2

$2$ is

2

$2$ .

\frac{2 π}{2}

$\frac{2 π}{2}$

Step 8.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 8.4.1

Cancel the common factor.

\frac{2 π}{2}

$\frac{2 π}{2}$

Step 8.4.2

Divide

π

$π$ by

1

$1$ .

π

$π$

π

$π$

π

$π$

Step 9

The period of the

\sin (2 x)

$\sin (2 x)$ function is

π

$π$ so values will repeat every

π

$π$ radians in both directions.

x = \frac{π}{12} + π n, \frac{5 π}{12} + π n

$x = \frac{π}{12} + π n, \frac{5 π}{12} + π n$ , for any integer

n

$n$

Step 10

Find the values of

n

$n$ that produce a value within the interval

[0, 2 π)

$[0, 2 π)$ .

Tap for more steps...

Step 10.1

Plug in

0

$0$ for

n

$n$ and simplify to see if the solution is contained in

[0, 2 π)

$[0, 2 π)$ .

Tap for more steps...

Step 10.1.1

Plug in

0

$0$ for

n

$n$ .

\frac{π}{12} + π (0)

$\frac{π}{12} + π (0)$

Step 10.1.2

Simplify.

Tap for more steps...

Step 10.1.2.1

Multiply

π

$π$ by

0

$0$ .

\frac{π}{12} + 0

$\frac{π}{12} + 0$

Step 10.1.2.2

Add

\frac{π}{12}

$\frac{π}{12}$ and

0

$0$ .

\frac{π}{12}

$\frac{π}{12}$

\frac{π}{12}

$\frac{π}{12}$

Step 10.1.3

The interval

[0, 2 π)

$[0, 2 π)$ contains

\frac{π}{12}

$\frac{π}{12}$ .

x = \frac{π}{12}

$x = \frac{π}{12}$

x = \frac{π}{12}

$x = \frac{π}{12}$

Step 10.2

Plug in

0

$0$ for

n

$n$ and simplify to see if the solution is contained in

[0, 2 π)

$[0, 2 π)$ .

Tap for more steps...

Step 10.2.1

Plug in

0

$0$ for

n

$n$ .

\frac{5 π}{12} + π (0)

$\frac{5 π}{12} + π (0)$

Step 10.2.2

Simplify.

Tap for more steps...

Step 10.2.2.1

Multiply

π

$π$ by

0

$0$ .

\frac{5 π}{12} + 0

$\frac{5 π}{12} + 0$

Step 10.2.2.2

Add

\frac{5 π}{12}

$\frac{5 π}{12}$ and

0

$0$ .

\frac{5 π}{12}

$\frac{5 π}{12}$

\frac{5 π}{12}

$\frac{5 π}{12}$

Step 10.2.3

The interval

[0, 2 π)

$[0, 2 π)$ contains

\frac{5 π}{12}

$\frac{5 π}{12}$ .

x = \frac{π}{12}, \frac{5 π}{12}

$x = \frac{π}{12}, \frac{5 π}{12}$

x = \frac{π}{12}, \frac{5 π}{12}

$x = \frac{π}{12}, \frac{5 π}{12}$

Step 10.3

Plug in

1

$1$ for

n

$n$ and simplify to see if the solution is contained in

[0, 2 π)

$[0, 2 π)$ .

Tap for more steps...

Step 10.3.1

Plug in

1

$1$ for

n

$n$ .

\frac{π}{12} + π (1)

$\frac{π}{12} + π (1)$

Step 10.3.2

Simplify.

Tap for more steps...

Step 10.3.2.1

Multiply

π

$π$ by

1

$1$ .

\frac{π}{12} + π

$\frac{π}{12} + π$

Step 10.3.2.2

To write

π

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

\frac{12}{12}

$\frac{12}{12}$ .

\frac{π}{12} + π \cdot \frac{12}{12}

$\frac{π}{12} + π \cdot \frac{12}{12}$

Step 10.3.2.3

Combine fractions.

Tap for more steps...

Step 10.3.2.3.1

Combine

π

$π$ and

\frac{12}{12}

$\frac{12}{12}$ .

\frac{π}{12} + \frac{π \cdot 12}{12}

$\frac{π}{12} + \frac{π \cdot 12}{12}$

Step 10.3.2.3.2

Combine the numerators over the common denominator.

\frac{π + π \cdot 12}{12}

$\frac{π + π \cdot 12}{12}$

\frac{π + π \cdot 12}{12}

$\frac{π + π \cdot 12}{12}$

Step 10.3.2.4

Simplify the numerator.

Tap for more steps...

Step 10.3.2.4.1

Move

12

$12$ to the left of

π

$π$ .

\frac{π + 12 \cdot π}{12}

$\frac{π + 12 \cdot π}{12}$

Step 10.3.2.4.2

Add

π

$π$ and

12 π

$12 π$ .

\frac{13 π}{12}

$\frac{13 π}{12}$

\frac{13 π}{12}

$\frac{13 π}{12}$

\frac{13 π}{12}

$\frac{13 π}{12}$

Step 10.3.3

The interval

[0, 2 π)

$[0, 2 π)$ contains

\frac{13 π}{12}

$\frac{13 π}{12}$ .

x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}

$x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}$

x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}

$x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}$

Step 10.4

Plug in

1

$1$ for

n

$n$ and simplify to see if the solution is contained in

[0, 2 π)

$[0, 2 π)$ .

Tap for more steps...

Step 10.4.1

Plug in

1

$1$ for

n

$n$ .

\frac{5 π}{12} + π (1)

$\frac{5 π}{12} + π (1)$

Step 10.4.2

Simplify.

Tap for more steps...

Step 10.4.2.1

Multiply

π

$π$ by

1

$1$ .

\frac{5 π}{12} + π

$\frac{5 π}{12} + π$

Step 10.4.2.2

To write

π

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

\frac{12}{12}

$\frac{12}{12}$ .

\frac{5 π}{12} + π \cdot \frac{12}{12}

$\frac{5 π}{12} + π \cdot \frac{12}{12}$

Step 10.4.2.3

Combine fractions.

Tap for more steps...

Step 10.4.2.3.1

Combine

π

$π$ and

\frac{12}{12}

$\frac{12}{12}$ .

\frac{5 π}{12} + \frac{π \cdot 12}{12}

$\frac{5 π}{12} + \frac{π \cdot 12}{12}$

Step 10.4.2.3.2

Combine the numerators over the common denominator.

\frac{5 π + π \cdot 12}{12}

$\frac{5 π + π \cdot 12}{12}$

\frac{5 π + π \cdot 12}{12}

$\frac{5 π + π \cdot 12}{12}$

Step 10.4.2.4

Simplify the numerator.

Tap for more steps...

Step 10.4.2.4.1

Move

12

$12$ to the left of

π

$π$ .

\frac{5 π + 12 \cdot π}{12}

$\frac{5 π + 12 \cdot π}{12}$

Step 10.4.2.4.2

Add

5 π

$5 π$ and

12 π

$12 π$ .

\frac{17 π}{12}

$\frac{17 π}{12}$

\frac{17 π}{12}

$\frac{17 π}{12}$

\frac{17 π}{12}

$\frac{17 π}{12}$

Step 10.4.3

The interval

[0, 2 π)

$[0, 2 π)$ contains

\frac{17 π}{12}

$\frac{17 π}{12}$ .

x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}, \frac{17 π}{12}

$x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}, \frac{17 π}{12}$

x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}, \frac{17 π}{12}

$x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}, \frac{17 π}{12}$

x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}, \frac{17 π}{12}

$x = \frac{π}{12}, \frac{5 π}{12}, \frac{13 π}{12}, \frac{17 π}{12}$