33 lines
839 B
Markdown
33 lines
839 B
Markdown
To solve for a double or half angle identity:
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1. Draw a triangle
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2. Choose an identity to use
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3. Substitute into formula
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# Double Angle Identities
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Sine:
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$$ \sin(2\theta) = 2\sin\theta\cos\theta $$
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Cosine:
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$$
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\begin{matrix}
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\cos(2\theta) = \cos^2\theta - \sin^2\theta\\
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= 1 - 2sin^2\theta\\
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= 2cos^2\theta - 1\\
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\end{matrix}
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$$
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Tan:
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$$ \tan(2\theta) = \dfrac{2\tan\theta}{1-\tan^2\theta}$$
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## Half Angle Identities
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Whether the output is positive or negative depends on what quadrant the output is in.
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Sine:
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$$ \sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1-\cos\theta}{2}} $$
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Cosine:
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$$ \cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos\theta}{2}} $$
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Tangent:
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$$
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\begin{matrix}
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\tan(\dfrac{\theta}{2}) = \pm\sqrt{\dfrac{1-\cos\theta}{1 + \cos\theta}}\\
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= \dfrac{\sin\theta}{1 + \cos\theta}\\
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= \dfrac{1 - cos\theta}{\sin\theta}
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\end{matrix}
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$$ |