Problem 112
Question
Use the appropriate values from Exercise 110 to answer each of the following. a. Is \(\cos \left(2 \cdot 30^{\circ}\right),\) or \(\cos 60^{\circ},\) equal to \(2 \cos 30^{\circ} ?\) b. Is \(\cos \left(2 \cdot 30^{\circ}\right),\) or \(\cos 60^{\circ},\) equal to \(\cos ^{2} 30^{\circ}-\sin ^{2} 30^{\circ} ?\)
Step-by-Step Solution
Verified Answer
\( \cos 60^{\circ} \) is not equal to \( 2 \cos 30^{\circ} \), but it is equal to \( \cos ^{2} 30^{\circ} - \sin ^{2} 30^{\circ} \)
1Step 1: Find the value of \( \cos 60^{\circ} \)
From the trigonometric functions tables, it is known that \( \cos 60^{\circ} = 0.5 \)
2Step 2: Find the value of \( 2 \cos 30^{\circ} \)
From the trigonometric functions tables, it is known that \( \cos 30^{\circ} = \sqrt{3}/2 \), thus \( 2 \cos 30^{\circ} = \sqrt{3} \). Therefore, \( \cos 60^{\circ} \) is not equal to \( 2 \cos 30^{\circ} \)
3Step 3: Find the value of \( \cos ^{2} 30^{\circ} - \sin ^{2} 30^{\circ} \)
From the trigonometric functions tables, it is known that \( \sin 30^{\circ} = 0.5 \), so \( \cos ^{2} 30^{\circ} - \sin ^{2} 30^{\circ} = (\sqrt{3}/2)^{2} - 0.5^{2} = 0.75 - 0.25 = 0.5 \). Thus, \( \cos 60^{\circ} \) is equal to \( \cos ^{2} 30^{\circ} - \sin ^{2} 30^{\circ} \)
Other exercises in this chapter
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