Find the exact value of the expression using the provided information.
-Find $\cos ( s - t )$ given that $\cos s = - \frac { 4 } { 5 }$, with $s$ in quadrant II, and $\cos t = \frac { 12 } { 13 }$, with $t$ in quadrant IV.
A) $- \frac { 33 } { 65 }$
B) $- \frac { 63 } { 65 }$
C) $\frac { 63 } { 65 }$
D) $\frac { 33 } { 65 }$

Question

Quizplus · Accepted Answer

The formula for $\cos(s - t)$ is $\cos s \cdot \cos t + \sin s \cdot \sin t$. 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, $\sin s = \sqrt{1 - \cos^2 s} = \sqrt{1 - \left(-\frac{4}{5}\right)^2} = \sqrt{1 - \frac{16}{25}} = \sqrt{\frac{9}{25}} = \frac{3}{5}$ (positive because $s$ is in quadrant II). For $t$ in quadrant IV, $\sin t = -\sqrt{1 - \cos^2 t} = -\sqrt{1 - \left(\frac{12}{13}\right)^2} = -\sqrt{1 - \frac{144}{169}} = -\sqrt{\frac{25}{169}} = -\frac{5}{13}$ (negative because $t$ is in quadrant IV). Substituting these into the formula gives $\cos(s - t) = \left(-\frac{4}{5}\right)\left(\frac{12}{13}\right) + \left(\frac{3}{5}\right)\left(-\frac{5}{13}\right) = -\frac{48}{65} - \frac{15}{65} = -\frac{63}{65}$.

Find the Exact Value of the Expression Using the Provided cos⁡(s−t)\cos ( s - t )cos(s−t)

Find the Exact Value of the Expression Using the Provided $\cos ( s - t )$