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Chapter 5: Problem 63

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$\pi / 8$$

Short Answer

Expert verified
The exact values for sine, cosine, and tangent of the angle \(\pi / 8\) are \(\sqrt{\frac{1-\sqrt{2}/2}{2}}\), \(\sqrt{\frac{1+\sqrt{2}/2}{2}}\) and \(\sqrt{\frac{1-\sqrt{2}/2}{2}} / \sqrt{\frac{1+\sqrt{2}/2}{2}}\) respectively.

Step by step solution

01

Apply the half angle identity for cosine

Start by finding the cosine of the half angle. The formula for the cosine of half of an angle is \(\cos(\theta/2) = \pm \sqrt{\frac{1+\cos(\theta)}{2}}\) . As \(\pi/4\) is the double of \(\pi / 8\), you should use \(\pi/4\) in place of \(\theta\). Also, \(\cos(\pi/4) = \sqrt{2} / 2\). So, \( \cos(\pi/8) = \sqrt{\frac{1+\sqrt{2}/2}{2}}\).
02

Apply the half angle identity for sine

The formula for the sine of half of an angle is \(\sin(\theta/2) = \pm \sqrt{\frac{1-\cos(\theta)}{2}}\). Use \(\pi/4\) for \(\theta\) and \(\cos(\pi/4) = \sqrt{2} / 2\). Then, \(\sin(\pi/8) = \sqrt{\frac{1-\sqrt{2}/2}{2}}\).
03

Divide sine by cosine to find tangent

The tangent of an angle is the sine of that angle divided by the cosine of that angle. So, \(\tan(\pi/8) = \sin(\pi/8) / \cos(\pi/8) = \sqrt{\frac{1-\sqrt{2}/2}{2}} / \sqrt{\frac{1+\sqrt{2}/2}{2}}\).

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