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Chapter 4: Problem 29

Find exact expressions for the indicated quantities, given that $$ \cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2} $$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived.] $$ \cos \left(-\frac{5 \pi}{12}\right) $$

Short Answer

Expert verified
The exact expression for the cosine of the given angle is: $$ \cos\left(-\frac{5\pi}{12}\right)= \frac{\sqrt{2+\sqrt{3}}}{4}+\frac{\sqrt{6-3\sqrt{2}}}{4} $$

Step by step solution

01

Identify Angle Composition

The first step is to express the given angle in terms of the provided angles. The angle \(-\frac{5\pi}{12}\) can be written as the sum of the angles \(\frac{-\pi}{12}\) and \(-\frac{\pi}{3}\).
02

Use the Cosine Difference Formula

Now, use the cosine difference formula to compute the cosine of the given angle. The cosine difference formula is given by: $$ \cos(a - b) = \cos a \cos b + \sin a \sin b $$ We will apply this formula to our given angle: $$ \cos \left(-\frac{5\pi}{12}\right) = \cos \left(\frac{-\pi}{12} - \frac{\pi}{3}\right) $$
03

Calculate Expressions for \(\cos(-\frac{\pi}{12})\) and \(\sin(-\frac{\pi}{12})\)

Since the cosine function is an even function, \(\cos(-x)=\cos(x)\). Hence: $$ \cos \left(-\frac{\pi}{12}\right) = \cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} $$ To find \(\sin \left(-\frac{\pi}{12}\right)\), we will use the fact that \(\sin(\frac{\pi}{2} - x) = \cos(x)\). Therefore: $$ \sin \left(-\frac{\pi}{12}\right)=\sin\left(\frac{\pi}{2} - \frac{\pi}{12}\right)= \sin \frac{\pi}{3} = \sqrt{\frac{3}{2}} $$
04

Calculate Expressions for \(\cos(-\frac{\pi}{3})\) and \(\sin(-\frac{\pi}{3})\)

To find \(\cos(-\frac{\pi}{3}),\) we will use the fact that \(\cos(\pi + x) = -\cos(x)\), so: $$ \cos \left(-\frac{\pi}{3}\right) = \cos\left(\pi + \frac{\pi}{3}\right) = -\cos\frac{\pi}{3} = \frac{1}{2} $$ Similar to finding \(\sin(-\frac{\pi}{12})\), to find \(\sin(-\frac{\pi}{3})\), we will use \(\sin(\frac{\pi}{3}-\varphi) = \cos(\varphi)\). Hence, $$ \sin \left(-\frac{\pi}{3}\right)=\sin\left(\frac{\pi}{3} - \theta\right)= \sin \frac{\pi}{8} = \frac{\sqrt{2-\sqrt{2}}}{2} $$
05

Plug in Values and Simplify

Now, substitute the values we found into the cosine difference formula. We have: $$ \cos \left(-\frac{5\pi}{12}\right)= \left(\frac{\sqrt{2+\sqrt{3}}}{2}\right) \left(\frac{1}{2}\right) + \left(\sqrt{\frac{3}{2}}\right) \left(\frac{\sqrt{2-\sqrt{2}}}{2}\right) $$ Simplify the expression: $$ \cos\left(-\frac{5\pi}{12}\right)= \frac{\sqrt{2+\sqrt{3}}}{4}+\frac{\sqrt{6-3\sqrt{2}}}{4} $$

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Most popular questions from this chapter

(a) Show that $$ x^{3}+x^{2} y+x y^{2}+y^{3}=\left(x^{2}+y^{2}\right)(x+y) $$ for all numbers \(x\) and \(y\). (b) Show that $$ \begin{aligned} \cos ^{3} \theta+\cos ^{2} \theta \sin \theta &+\cos \theta \sin ^{2} \theta+\sin ^{3} \theta \\ &=\cos \theta+\sin \theta \end{aligned} $$ for every number \(\theta\).

Find exact expressions for the indicated quantities, given that $$ \cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2} $$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived.] $$ \cos \left(-\frac{5 \pi}{12}\right) $$

Find exact expressions for the indicated quantities, given that $$ \cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2} $$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived.] $$ \tan \left(-\frac{5 \pi}{12}\right) $$

Show that $$ \cos (\pi-\theta)=-\cos \theta $$ for every angle \(\theta\).

Give exact values for the quantities. Do not use a calculator for any of these exercises-otherwise you will likely get decimal approximations for some solutions rather than exact answers. More importantly, good understanding will come from working these exercises by hand. (a) \(\cos 360045^{\circ}\) (b) \(\sin 360045^{\circ}\)
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