Trigonometry Examples

$\sin (\frac{5 π}{6} + x)$

Step 1

Apply the sum of angles identity.

$\sin (\frac{5 π}{6}) \cos (x) + \cos (\frac{5 π}{6}) \sin (x)$

Step 2

Simplify each term.

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Step 2.1

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

$\sin (\frac{π}{6}) \cos (x) + \cos (\frac{5 π}{6}) \sin (x)$

Step 2.2

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

$\frac{1}{2} \cos (x) + \cos (\frac{5 π}{6}) \sin (x)$

Step 2.3

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

$\frac{\cos (x)}{2} + \cos (\frac{5 π}{6}) \sin (x)$

Step 2.4

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 second quadrant.

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

Step 2.5

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

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

Step 2.6

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

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