Use identities to write each expression as a function of .
-Find $\cos ( s - t )$ given that $\cos s = \frac { \sqrt { 10 } } { 10 }$, with $s$ in quadrant $I$, and $\sin t = - \frac { \sqrt { 3 } } { 2 }$, with $t$ in quadrant IV.
A) $\frac { \sqrt { 10 } - 3 \sqrt { 30 } } { 20 }$
B) $- \frac { \sqrt { 10 } } { 20 }$
C) $\frac { \sqrt { 10 } - \sqrt { 30 } } { 19 }$
D) $\frac { \sqrt { 10 } + 3 \sqrt { 30 } } { 20 }$

Question

Quizplus · Accepted Answer

The correct expression for $\cos(s - t)$ can be found using the identity $\cos(s - t) = \cos s \cos t + \sin s \sin t$. Given $\cos s = \frac{\sqrt{10}}{10}$ and $\sin t = -\frac{\sqrt{3}}{2}$ with $s$ in quadrant I and $t$ in quadrant IV, we find $\sin s = \sqrt{1 - \cos^2 s} = \sqrt{1 - \frac{10}{100}} = \sqrt{\frac{90}{100}} = \frac{3\sqrt{10}}{10}$ and $\cos t = \sqrt{1 - \sin^2 t} = \sqrt{1 - \frac{3}{4}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$. Substituting these values into the identity gives $\cos(s - t) = \frac{\sqrt{10}}{10} \cdot \frac{1}{2} + \frac{3\sqrt{10}}{10} \cdot \left(-\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{10}}{20} - \frac{3\sqrt{30}}{20} = \frac{\sqrt{10} - 3\sqrt{30}}{20}$.

Use Identities to Write Each Expression as a Function of cos⁡(s−t)\cos ( s - t )cos(s−t)

Use Identities to Write Each Expression as a Function of $\cos ( s - t )$