Trigonometry Examples

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

Step 1

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

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Step 1.1

Remove parentheses.

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

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.

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

Step 1.2.2

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 1.2.3

Cancel the common factor of $2$ .

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Step 1.2.3.1

Factor $2$ out of $6$ .

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

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