Questions: Calculate an exact answer using a sum or difference formula. sin 165°

Solution Steps

To calculate \(\sin 165^{\circ}\), we can use the sum formula for sine. We know that \(165^{\circ} = 120^{\circ} + 45^{\circ}\). Using the sine sum formula \(\sin(a + b) = \sin a \cos b + \cos a \sin b\), we can find the exact value.

We will use the sum formula for sine: \[ \sin(165^{\circ}) = \sin(120^{\circ} + 45^{\circ}) = \sin 120^{\circ} \cos 45^{\circ} + \cos 120^{\circ} \sin 45^{\circ} \] We know the exact values: \[ \sin 120^{\circ} = \frac{\sqrt{3}}{2}, \quad \cos 120^{\circ} = -\frac{1}{2}, \quad \sin 45^{\circ} = \frac{\sqrt{2}}{2}, \quad \cos 45^{\circ} = \frac{\sqrt{2}}{2} \] Substitute these values into the formula to get the exact answer.

We start by identifying the values needed for the sine sum formula. We have: \[ \sin 120^{\circ} = \frac{\sqrt{3}}{2} \quad \text{and} \quad \cos 120^{\circ} = -\frac{1}{2} \] For \(45^{\circ}\): \[ \sin 45^{\circ} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \]

Using the sine sum formula: \[ \sin(165^{\circ}) = \sin(120^{\circ} + 45^{\circ}) = \sin 120^{\circ} \cos 45^{\circ} + \cos 120^{\circ} \sin 45^{\circ} \] Substituting the known values: \[ \sin(165^{\circ}) = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) + \left(-\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) \]

Calculating each term: \[ \sin(165^{\circ}) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} \] Combining the terms: \[ \sin(165^{\circ}) = \frac{\sqrt{6} - \sqrt{2}}{4} \]

Calculating the numerical value: \[ \sin(165^{\circ}) \approx 0.2588 \]

Final Answer