3.6 Transformation Formulas (Product <-> Sum)

Value of $\sin 60^\circ = \frac{\sqrt{3}}{2}$ .
Expression: $\frac{\sqrt{3}}{2} [ \sin 20^\circ \sin 40^\circ \sin 80^\circ ]$ .
Multiply/Divide by 2: $\frac{\sqrt{3}}{4} [ (2 \sin 80 \sin 20) \sin 40 ]$ .
Use $2\sin A \sin B = \cos(A-B) - \cos(A+B)$ : $2 \sin 80 \sin 20 = \cos 60 - \cos 100 = \frac{1}{2} - \cos 100$ .
Now: $\frac{\sqrt{3}}{4} [ \frac{1}{2} \sin 40 - \cos 100 \sin 40 ]$ .
$2 \cos 100 \sin 40 = \sin 140 - \sin 60 = \sin(180-40) - \frac{\sqrt{3}}{2} = \sin 40 - \frac{\sqrt{3}}{2}$ .
Substitute back: The terms cancel nicely to result in $\frac{3}{16}$ .

Operation

Angle

Unit

Standard Values:

sin 0 = 0, sin 30 = 0.5, sin 45 = 0.707, sin 60 = 0.866, sin 90 = 1

cos 0 = 1, cos 30 = 0.866, cos 45 = 0.707, cos 60 = 0.5, cos 90 = 0

Select an operation and enter values.