Trigonometry Examples

$\sin (\frac{7 π}{6} + x) - \cos (\frac{2 π}{3} + x) = 0$

Step 1

Start on the left side.

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

Step 2

Apply the sum of angles identity.

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

Step 3

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

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

Step 4

Simplify the expression.

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Step 4.1

Simplify each term.

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Step 4.1.1

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.

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

Step 4.1.2

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

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

Step 4.1.3

Combine $\cos (x)$ and $\frac{1}{2}$ .

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

Step 4.1.4

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

$- \frac{\cos (x)}{2} - \cos (\frac{π}{6}) \sin (x) - (\cos (\frac{2 π}{3}) \cos (x) - \sin (\frac{2 π}{3}) \sin (x))$

Step 4.1.5

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$- \frac{\cos (x)}{2} - \frac{\sqrt{3}}{2} \sin (x) - (\cos (\frac{2 π}{3}) \cos (x) - \sin (\frac{2 π}{3}) \sin (x))$

Step 4.1.6

Combine $\sin (x)$ and $\frac{\sqrt{3}}{2}$ .

$- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} - (\cos (\frac{2 π}{3}) \cos (x) - \sin (\frac{2 π}{3}) \sin (x))$

Step 4.1.7

Simplify each term.

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Step 4.1.7.1

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

$- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} - (- \cos (\frac{π}{3}) \cos (x) - \sin (\frac{2 π}{3}) \sin (x))$

Step 4.1.7.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} - (- \frac{1}{2} \cos (x) - \sin (\frac{2 π}{3}) \sin (x))$

Step 4.1.7.3

Combine $\cos (x)$ and $\frac{1}{2}$ .

$- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} - (- \frac{\cos (x)}{2} - \sin (\frac{2 π}{3}) \sin (x))$

Step 4.1.7.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} - (- \frac{\cos (x)}{2} - \sin (\frac{π}{3}) \sin (x))$

Step 4.1.7.5

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} - (- \frac{\cos (x)}{2} - \frac{\sqrt{3}}{2} \sin (x))$

Step 4.1.7.6

Combine $\sin (x)$ and $\frac{\sqrt{3}}{2}$ .

$- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} - (- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2})$

Step 4.1.8

Apply the distributive property.

$- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} - - \frac{\cos (x)}{2} - - \frac{\sin (x) \sqrt{3}}{2}$

Step 4.1.9

Multiply $- - \frac{\cos (x)}{2}$ .

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Step 4.1.9.1

Multiply $- 1$ by $- 1$ .

$- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + 1 \frac{\cos (x)}{2} - - \frac{\sin (x) \sqrt{3}}{2}$

Step 4.1.9.2

Multiply $\frac{\cos (x)}{2}$ by $1$ .

$- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + \frac{\cos (x)}{2} - - \frac{\sin (x) \sqrt{3}}{2}$

Step 4.1.10

Multiply $- - \frac{\sin (x) \sqrt{3}}{2}$ .

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Step 4.1.10.1

Multiply $- 1$ by $- 1$ .

$- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + \frac{\cos (x)}{2} + 1 \frac{\sin (x) \sqrt{3}}{2}$

Step 4.1.10.2

Multiply $\frac{\sin (x) \sqrt{3}}{2}$ by $1$ .

$- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + \frac{\cos (x)}{2} + \frac{\sin (x) \sqrt{3}}{2}$

Step 4.2

Combine the opposite terms in $- \frac{\cos (x)}{2} - \frac{\sin (x) \sqrt{3}}{2} + \frac{\cos (x)}{2} + \frac{\sin (x) \sqrt{3}}{2}$ .

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Step 4.2.1

Add $- \frac{\cos (x)}{2}$ and $\frac{\cos (x)}{2}$ .

$0 - \frac{\sin (x) \sqrt{3}}{2} + \frac{\sin (x) \sqrt{3}}{2}$

Step 4.2.2

Subtract $\frac{\sin (x) \sqrt{3}}{2}$ from $0$ .

$- \frac{\sin (x) \sqrt{3}}{2} + \frac{\sin (x) \sqrt{3}}{2}$

Step 4.3

Combine the numerators over the common denominator.

$\frac{- \sin (x) \sqrt{3} + \sin (x) \sqrt{3}}{2}$

Step 4.4

Add $- \sin (x) \sqrt{3}$ and $\sin (x) \sqrt{3}$ .

$\frac{0}{2}$

Step 4.5

Divide $0$ by $2$ .

$0$

Step 5

Because the two sides have been shown to be equivalent, the equation is an identity.

$\sin (\frac{7 π}{6} + x) - \cos (\frac{2 π}{3} + x) = 0$ is an identity