Determine all solutions of the equation in radians.
-Find $\cos ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 4 } { 5 }$, with $\mathrm { s }$ in quadrant II, and $\cos \mathrm { t } = \frac { 12 } { 13 }$, with $\mathrm { t }$ in quadrant IV.
A) $- \frac { 63 } { 65 }$
B) $- \frac { 33 } { 65 }$
C) $\frac { 63 } { 65 }$
D) $-$

Question

Quizplus · Accepted Answer

The cosine of a difference, $\cos(s - t)$, can be found using the formula $\cos(s - t) = \cos s \cos t + \sin s \sin t$. Given $\cos s = -\frac{4}{5}$ and $\cos t = \frac{12}{13}$, we find $\sin s$ and $\sin t$ using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$. For $s$ in quadrant II, $\sin s = \sqrt{1 - \cos^2 s} = \sqrt{1 - \left(-\frac{4}{5}\right)^2} = \frac{3}{5}$ (positive because $s$ is in quadrant II). For $t$ in quadrant IV, $\sin t = -\sqrt{1 - \cos^2 t} = -\sqrt{1 - \left(\frac{12}{13}\right)^2} = -\frac{5}{13}$ (negative because $t$ is in quadrant IV). Plugging these into the formula gives $\cos(s - t) = \left(-\frac{4}{5}\right)\left(\frac{12}{13}\right) + \left(\frac{3}{5}\right)\left(-\frac{5}{13}\right) = -\frac{48}{65} - \frac{15}{65} = -\frac{63}{65}$.

Determine All Solutions of the Equation in Radians cos⁡(s−t)\cos ( \mathrm { s } - \mathrm { t } )cos(s−t)

Determine All Solutions of the Equation in Radians $\cos ( \mathrm { s } - \mathrm { t } )$