What is the exact value of \( \sin 165^{\circ}+\sin 75^{\circ} \) ? \( -\frac{\sqrt{3}}{2} \) \( -\frac{\sqrt{6}}{2} \)

To find the exact value of \( \sin 165^{\circ}+\sin 75^{\circ} \), we can use the sine addition formula and some properties of sine. First, we know that: \[ \sin 165^{\circ} = \sin(180^{\circ} - 15^{\circ}) = \sin 15^{\circ} \] and \[ \sin 75^{\circ} = \sin(90^{\circ} + 15^{\circ}) = \cos 15^{\circ} \] So, we can re-write the expression as: \[ \sin 165^{\circ} + \sin 75^{\circ} = \sin 15^{\circ} + \cos 15^{\circ} \] Using the identity \(\sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right)\), we substitute \(x = 15^{\circ}\): \[ \sin 15^{\circ} + \cos 15^{\circ} = \sqrt{2} \sin\left(15^{\circ} + 45^{\circ}\right) = \sqrt{2} \sin 60^{\circ} = \sqrt{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2} \] So, the exact value of \( \sin 165^{\circ}+\sin 75^{\circ} \) is \( \frac{\sqrt{6}}{2} \). Now, let’s explore a common mistake—many might overlook co-function identities! Remember that \( \sin(90^\circ - x) = \cos(x) \). This relationship can help simplify calculations involving sines and cosines. Additionally, for further exploration, try using the unit circle for visual concepts! For example, the sine of an angle corresponds to the y-coordinate of a point on the unit circle, which visually demonstrates sine values as you trace through different angles. It’s fun to see how adding angles affects values!