Question

Find the exact value of sin 165° using expansion of sin (a + b).

• A. sin(165°) = sin(120°)cos(45°) + cos(120°)sin(45°)
• B. sin(165°) = sin(180°)cos(15°) - cos(180°)sin(15°)
• C. sin(165°) = sin(120° + 45°) = sin(120°)cos(45°) + cos(120°)sin(45°)
• D. sin(165°) = sin(135° + 30°) = sin(135°)cos(30°) + cos(135°)sin(30°)

Answer 1

Answer: (C)

sin(165°) = sin(120° + 45°) = sin(120°)cos(45°) + cos(120°)sin(45°)

Answer 2

To find the exact value of sin 165°, we can use the sine addition formula: $\sin(a+b) = \sin a \cos b + \cos a \sin b$. We express 165° as a sum of two standard angles, for example, $120^\circ + 45^\circ$.

Applying the formula: $\sin(165^\circ) = \sin(120^\circ + 45^\circ) = \sin(120^\circ)\cos(45^\circ) + \cos(120^\circ)\sin(45^\circ)$

Now, we substitute the known values of these trigonometric functions: $\sin(120^\circ) = \frac{\sqrt{3}}{2}$ $\cos(120^\circ) = -\frac{1}{2}$ $\sin(45^\circ) = \frac{\sqrt{2}}{2}$ $\cos(45^\circ) = \frac{\sqrt{2}}{2}$

Substituting these values into the equation: $\sin(165^\circ) = \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(-\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right)$ $\sin(165^\circ) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$ $\sin(165^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}$

Therefore, the exact value of $\sin 165^\circ$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$. The option that correctly represents the application of the sine addition formula for sin 165° is $\sin(165^\circ) = \sin(120^\circ + 45^\circ) = \sin(120^\circ)\cos(45^\circ) + \cos(120^\circ)\sin(45^\circ)$.