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.

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

Step 1.2.3.3

Rewrite the expression.

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

Step 1.2.4

Subtract full rotations of

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

$0$ and less than

$2 π$ .

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

Step 1.2.5

The exact value of

$\sin (\frac{π}{3})$ is

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

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

Step 2

Simplify

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

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine

$\sin (2 x)$ and

$\frac{7 π}{6}$ .

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

Step 2.1.2

Move

$7$ to the left of

$\sin (2 x)$ .

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

Step 2.2

Reorder factors in

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

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

Step 3

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

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

$n$

Step 4