Trigonometry Examples

sin (\frac{7 π}{6}) + sin (2 (\frac{7 π}{6})) = (sin (x) + sin (2 x) (\frac{7 π}{6}))

$\sin (\frac{7 π}{6}) + \sin (2 (\frac{7 π}{6})) = (\sin (x) + \sin (2 x) (\frac{7 π}{6}))$

Step 1

Simplify

sin (\frac{7 π}{6}) + sin (2 (\frac{7 π}{6}))

$\sin (\frac{7 π}{6}) + \sin (2 (\frac{7 π}{6}))$ .

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Step 1.1

Remove parentheses.

sin (\frac{7 π}{6}) + sin (2 (\frac{7 π}{6})) = sin (x) + sin (2 x) (\frac{7 π}{6})

$\sin (\frac{7 π}{6}) + \sin (2 (\frac{7 π}{6})) = \sin (x) + \sin (2 x) (\frac{7 π}{6})$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

- sin (\frac{π}{6}) + sin (2 (\frac{7 π}{6})) = sin (x) + sin (2 x) (\frac{7 π}{6})

$- \sin (\frac{π}{6}) + \sin (2 (\frac{7 π}{6})) = \sin (x) + \sin (2 x) (\frac{7 π}{6})$

Step 1.2.2

The exact value of

sin (\frac{π}{6})

$\sin (\frac{π}{6})$ is

\frac{1}{2}

$\frac{1}{2}$ .

- \frac{1}{2} + sin (2 (\frac{7 π}{6})) = sin (x) + sin (2 x) (\frac{7 π}{6})

$- \frac{1}{2} + \sin (2 (\frac{7 π}{6})) = \sin (x) + \sin (2 x) (\frac{7 π}{6})$

Step 1.2.3

Cancel the common factor of

2

$2$ .

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Step 1.2.3.1

Factor

2

$2$ out of

6

$6$ .

- \frac{1}{2} + sin (2 \frac{7 π}{2 (3)}) = sin (x) + sin (2 x) (\frac{7 π}{6})

$- \frac{1}{2} + \sin (2 \frac{7 π}{2 (3)}) = \sin (x) + \sin (2 x) (\frac{7 π}{6})$

Step 1.2.3.2

Cancel the common factor.

- \frac{1}{2} + sin (2 \frac{7 π}{2 \cdot 3}) = sin (x) + sin (2 x) (\frac{7 π}{6})

$- \frac{1}{2} + \sin (2 \frac{7 π}{2 \cdot 3}) = \sin (x) + \sin (2 x) (\frac{7 π}{6})$

Step 1.2.3.3

Rewrite the expression.

- \frac{1}{2} + sin (\frac{7 π}{3}) = sin (x) + sin (2 x) (\frac{7 π}{6})

$- \frac{1}{2} + \sin (\frac{7 π}{3}) = \sin (x) + \sin (2 x) (\frac{7 π}{6})$

- \frac{1}{2} + sin (\frac{7 π}{3}) = sin (x) + sin (2 x) (\frac{7 π}{6})

$- \frac{1}{2} + \sin (\frac{7 π}{3}) = \sin (x) + \sin (2 x) (\frac{7 π}{6})$

Step 1.2.4

Subtract full rotations of

2 π

$2 π$ until the angle is greater than or equal to

0

$0$ and less than

2 π

$2 π$ .

- \frac{1}{2} + sin (\frac{π}{3}) = sin (x) + sin (2 x) (\frac{7 π}{6})

$- \frac{1}{2} + \sin (\frac{π}{3}) = \sin (x) + \sin (2 x) (\frac{7 π}{6})$

Step 1.2.5

The exact value of

sin (\frac{π}{3})

$\sin (\frac{π}{3})$ is

\frac{\sqrt{3}}{2}

$\frac{\sqrt{3}}{2}$ .

- \frac{1}{2} + \frac{\sqrt{3}}{2} = sin (x) + sin (2 x) (\frac{7 π}{6})

$- \frac{1}{2} + \frac{\sqrt{3}}{2} = \sin (x) + \sin (2 x) (\frac{7 π}{6})$

- \frac{1}{2} + \frac{\sqrt{3}}{2} = sin (x) + sin (2 x) (\frac{7 π}{6})

$- \frac{1}{2} + \frac{\sqrt{3}}{2} = \sin (x) + \sin (2 x) (\frac{7 π}{6})$

- \frac{1}{2} + \frac{\sqrt{3}}{2} = sin (x) + sin (2 x) (\frac{7 π}{6})

$- \frac{1}{2} + \frac{\sqrt{3}}{2} = \sin (x) + \sin (2 x) (\frac{7 π}{6})$

Step 2

Simplify

sin (x) + sin (2 x) (\frac{7 π}{6})

$\sin (x) + \sin (2 x) (\frac{7 π}{6})$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine

sin (2 x)

$\sin (2 x)$ and

\frac{7 π}{6}

$\frac{7 π}{6}$ .

- \frac{1}{2} + \frac{\sqrt{3}}{2} = sin (x) + \frac{sin (2 x) (7 π)}{6}

$- \frac{1}{2} + \frac{\sqrt{3}}{2} = \sin (x) + \frac{\sin (2 x) (7 π)}{6}$

Step 2.1.2

Move

7

$7$ to the left of

sin (2 x)

$\sin (2 x)$ .

- \frac{1}{2} + \frac{\sqrt{3}}{2} = sin (x) + \frac{7 sin (2 x) π}{6}

$- \frac{1}{2} + \frac{\sqrt{3}}{2} = \sin (x) + \frac{7 \sin (2 x) π}{6}$

- \frac{1}{2} + \frac{\sqrt{3}}{2} = sin (x) + \frac{7 sin (2 x) π}{6}

$- \frac{1}{2} + \frac{\sqrt{3}}{2} = \sin (x) + \frac{7 \sin (2 x) π}{6}$

Step 2.2

Reorder factors in

sin (x) + \frac{7 sin (2 x) π}{6}

$\sin (x) + \frac{7 \sin (2 x) π}{6}$ .

- \frac{1}{2} + \frac{\sqrt{3}}{2} = sin (x) + \frac{7 π sin (2 x)}{6}

$- \frac{1}{2} + \frac{\sqrt{3}}{2} = \sin (x) + \frac{7 π \sin (2 x)}{6}$

- \frac{1}{2} + \frac{\sqrt{3}}{2} = sin (x) + \frac{7 π sin (2 x)}{6}

$- \frac{1}{2} + \frac{\sqrt{3}}{2} = \sin (x) + \frac{7 π \sin (2 x)}{6}$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

x \approx 0.04399022 + 2 π n, 1.6572027 + 2 π n, 3.1995582 + 2 π n, 4.52402682 + 2 π n

$x \approx 0.04399022 + 2 π n, 1.6572027 + 2 π n, 3.1995582 + 2 π n, 4.52402682 + 2 π n$ , for any integer

n

$n$

Step 4