Simplify sin(-60)cos(40)+cos(-60)sin(40)

$\sin (- 60) \cos (40) + \cos (- 60) \sin (40)$

Step 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 fourth quadrant.

$- \sin (60) \cos (40) + \cos (- 60) \sin (40)$

Step 2

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$- \frac{\sqrt{3}}{2} \cos (40) + \cos (- 60) \sin (40)$

Step 3

Combine $\cos (40)$ and $\frac{\sqrt{3}}{2}$ .

$- \frac{\cos (40) \sqrt{3}}{2} + \cos (- 60) \sin (40)$

Step 4

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

$- \frac{\cos (40) \sqrt{3}}{2} + \cos (60) \sin (40)$

Step 5

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$- \frac{\cos (40) \sqrt{3}}{2} + \frac{1}{2} \sin (40)$

Step 6

Combine $\frac{1}{2}$ and $\sin (40)$ .

$- \frac{\cos (40) \sqrt{3}}{2} + \frac{\sin (40)}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{\cos (40) \sqrt{3}}{2} + \frac{\sin (40)}{2}$

Decimal Form:

$- 0.34202014 \dots$

Simplify sin(-60)cos(40)+cos(-60)sin(40)

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