Find the exact value of sin(-165^circ ) by using a sum or difference formula. square

To find the exact value of $\sin(-165^{\circ})$, we can use the sum or difference formula for sine. Let's use the difference formula:<br /><br />$\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)$<br /><br />We can express $-165^{\circ}$ as $-180^{\circ} + 15^{\circ}$.<br /><br />So, $\sin(-165^{\circ}) = \sin(-180^{\circ} + 15^{\circ})$<br /><br />Using the difference formula with $a = -180^{\circ}$ and $b = 15^{\circ}$:<br /><br />$\sin(-165^{\circ}) = \sin(-180^{\circ})\cos(15^{\circ}) - \cos(-180^{\circ})\sin(15^{\circ})$<br /><br />We know that $\sin(-180^{\circ}) = -\sin(180^{\circ}) = 0$ and $\cos(-180^{\circ}) = -\cos(180^{\circ}) = 1$.<br /><br />Therefore:<br /><br />$\sin(-165^{\circ}) = 0 \cdot \cos(15^{\circ}) - 1 \cdot \sin(15^{\circ})$<br /><br />$\sin(-165^{\circ}) = -\sin(15^{\circ})$<br /><br />The exact value of $\sin(-165^{\circ})$ is $-\sin(15^{\circ})$.