Use identities to fill in the blank with the appropriate trigonometric function name.
-Find $\sin ( \mathrm { s } - \mathrm { t } )$ given that $\sin \mathrm { s } = - \frac { 1 } { 2 }$, with $\mathrm { s }$ in quadrant $\mathrm { IV }$, and $\sin \mathrm { t } = \frac { 1 } { 4 }$, with $\mathrm { t }$ in quadrant II.
A) $\frac { \sqrt { 15 } + \sqrt { 3 } } { 8 }$
B) $\frac { - \sqrt { 15 } - \sqrt { 3 } } { 8 }$
C) $\frac { - \sqrt { 15 } + \sqrt { 3 } } { 8 }$
D) $\frac { \sqrt { 15 } - \sqrt { 3 } } { 8 }$

Question

Quizplus · Accepted Answer

Using the sine difference identity $\sin(s - t) = \sin s \cos t - \cos s \sin t$, and knowing $\sin s = -\frac{1}{2}$ and $\sin t = \frac{1}{4}$, we find $\cos s = \sqrt{1 - \sin^2 s} = \frac{\sqrt{3}}{2}$ (positive in quadrant IV) and $\cos t = -\sqrt{1 - \sin^2 t} = -\frac{\sqrt{15}}{4}$ (negative in quadrant II). Substituting these values gives $\sin(s - t) = -\frac{1}{2}(-\frac{\sqrt{15}}{4}) - \frac{\sqrt{3}}{2}(\frac{1}{4}) = \frac{\sqrt{15}}{8} - \frac{\sqrt{3}}{8} = \frac{\sqrt{15} - \sqrt{3}}{8}$.

Use Identities to Fill in the Blank with the Appropriate sin⁡(s−t)\sin ( \mathrm { s } - \mathrm { t } )sin(s−t)

Use Identities to Fill in the Blank with the Appropriate $\sin ( \mathrm { s } - \mathrm { t } )$