Unformatted text preview: = 1 − 2 sin2 (θ)
• sin(2θ) = 2 sin(θ) cos(θ)
• tan(2θ) = 2 tan(θ)
1 − tan2 (θ) The three diﬀerent forms for cos(2θ) can be explained by our ability to ‘exchange’ squares of cosine
and sine via the Pythagorean Identity cos2 (θ) + sin2 (θ) = 1 and we leave the details to the reader.
It is interesting to note that to determine the value of cos(2θ), only one piece of information is
required: either cos(θ) or sin(θ). To determine sin(2θ), however, it appears that we must know
both sin(θ) and cos(θ). In the next example, we show how we can ﬁnd sin(2θ) knowing just one
piece of information, namely tan(θ).
1. Suppose P (−3, 4) lies on the terminal side of θ when θ is plotted in standard position. Find
cos(2θ) and sin(2θ) and determine the quadrant in which the terminal side of the angle 2θ
lies when it is plotted in standard position.
2. If sin(θ) = x for − π ≤ θ ≤ π , ﬁnd an expression for sin(2θ) in terms of x.
3. Verify the identity: sin(2θ) = 2 tan(θ)
1 + tan2 (θ) 4. Express cos(...
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