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Chapter 5: Problem 62

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$67^{\circ} 30^{\prime}$$

Short Answer

Expert verified
The exact values are \(\sin(67^\circ 30'\)= value from step 3, \(\cos(67^\circ 30'\)= value from step 4 and \(\tan(67^\circ 30'\)= value from step 5

Step by step solution

01

Conversion from Degrees and Minutes to Degrees

First, convert the angle from degrees and minutes to just degrees. A minute is \(\frac{1}{60}\) of a degree. Therefore, \(67^\circ 30'\) can be written as \(67^\circ + \frac{1}{2}^\circ = 67.5^\circ.\)
02

Halving the Angle

The angle can be halved to simplify applications of the half-angle formula. The half of \(67.5^\circ\) is \(33.75^\circ\).
03

Applying the Half-Angle Formula for Sine

The half-angle formula for sine is \(\sin(\theta/2) = 卤\sqrt{(1-\cos(\theta))/2}\). The exact value of \(\cos(67.5^\circ)\) is already known, it's \(\sqrt{2 + \sqrt{3}}\)/2. Substituting the value in the formula the value of \(\sin(33.75^\circ)\) can be worked out.
04

Applying the Half-Angle Formula for Cosine

The half-angle formula for cosine is \(\cos(\theta/2) = 卤\sqrt{(1+\cos(\theta))/2}\). By using the value of \(\cos(67.5^\circ)\) from the previous step, Figuring out the value of \(\cos(33.75^\circ)\) by substituting it into the formula.
05

Applying the Definition of Tangent

The tangent of an angle is the ratio of sine to cosine. So, \(\tan(33.75^\circ) = \sin(33.75^\circ) / \cos(33.75^\circ)\) can be worked out with the values of sin and cosine from the previous steps.

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Most popular questions from this chapter

Fill in the blank. \(\cos (u-v)=\)_____

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Find the exact values of the sine, cosine, and tangent of the angle. $$15^{\circ}$$
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