Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 5 - Section 5.2 - Sum and Difference Formulas - Exercise Set - Page 669: 20


The exact value of $\cos 105{}^\circ $ is $\frac{1-\sqrt{3}}{2\sqrt{2}}$.

Work Step by Step

Rewrite the expression for cosine of $75{}^\circ $ as the sum of two angles as, $\cos 105{}^\circ =\cos \left( 60{}^\circ +45{}^\circ \right)$ Use the sum formula of cosines and evaluate the modified expression as, $\cos \left( 60{}^\circ +45{}^\circ \right)=\cos 60{}^\circ \cos 45{}^\circ -\sin 60{}^\circ \sin 45{}^\circ $ Substitute the values, $\cos 45{}^\circ =\frac{1}{\sqrt{2}},\text{ }\cos 60{}^\circ =\frac{1}{2},\text{ }\sin 45{}^\circ =\frac{1}{\sqrt{2}},\text{ and }\sin 60{}^\circ =\frac{\sqrt{3}}{2}$. $\begin{align} & \cos \left( 60{}^\circ +45{}^\circ \right)=\left( \left( \frac{1}{2}\times \frac{1}{\sqrt{2}} \right)-\left( \frac{\sqrt{3}}{2}\times \frac{1}{\sqrt{2}} \right) \right) \\ & =\frac{1}{2\sqrt{2}}-\frac{\sqrt{3}}{2\sqrt{2}} \\ & =\frac{1-\sqrt{3}}{2\sqrt{2}} \end{align}$ Hence, the exact value of $\cos 105{}^\circ $ is equivalent to $\frac{1-\sqrt{3}}{2\sqrt{2}}$.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.