Problem 91 - Algebra and Trigonometry

Question

How can there be three forms of the double-angle formula for \(\cos 2 \theta ?\)

Step-by-Step Solution

Answer

The three forms of the double-angle formula for \(cos 2 \theta\) can be derived as follows: \(cos 2 \theta = 1 - 2 sin^{2} \theta\), \(cos 2 \theta = 2 cos^{2} \theta - 1\), and by substituting \(cos^{2} \theta = \frac{1}{2} + \frac{0.5 cos 2 \theta}{2}\) into the second form, we get the third form: \(cos 2 \theta = 1 - 2 sin^{2} \theta\).

Step 1: Express \(cos 2 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\)

The Cosine Double-Angle Formula is: \[cos 2 \theta = 1 - 2 \sin^{2} \theta\] This first form connects \(cos 2 \theta\) with \(\sin \theta\).

Step 2: Substitute \(\sin^{2} \theta\) with \(1 - \cos^{2} \theta\)

By substituting \(\sin^{2} \theta = 1 - \cos^{2} \theta\) into the first form: \[cos 2 \theta = 1 - 2 (1 - \cos^{2} \theta) = 2 \cos^{2} \theta - 1\] We obtained the second form that relates \(cos 2 \theta\) with \(\cos \theta\).

Step 3: Substitute \(\cos^{2} \theta\) with \(\frac{1}{2} + \frac{0.5 \cos 2 \theta}{2}\)

By substituting \(\cos^{2} \theta = \frac{1}{2} + \frac{0.5 \cos 2 \theta}{2}\) into the second form: \[cos 2 \theta = 2\left(\frac{1}{2} + \frac{0.5 \cos 2 \theta}{2}\right) - 1 = 1 - 2 \sin^{2} \theta\] We obtained the third form that relates \(cos 2 \theta\) with \(\sin \theta\) again.