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Math Help - Compound Angles

  1. #1
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    Question Compound Angles

    hi people i have a question on compound angles which im finding quite annoying could someone show me how to solve the following questions:

    express the following

    1. 6sinwt - 8coswt in the form of: R sin(Θ - a)

    2. 10sinΘ + 12sin(Θ+30) in the form of: R sin (Θ + a)
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  2. #2
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    Re: Compound Angles

    You know that 3^2+4^2 = 5^2, so 6^2+8^2 = 10^2. Hence, for #1, factor out 10:

    10\left(\dfrac{3}{5} \sin (wt) - \dfrac{4}{5} \cos(wt) \right)

    There exists an angle a such that \cos a = \dfrac{3}{5} and \sin a = \dfrac{4}{5} (it is the angle adjacent to the sides of length 3 and 5 in a 3-4-5 right triangle).

    So, you have 10(\sin(wt)\cos(a) - \cos(wt)\sin(a)) = 10\sin(wt - a).

    Do something similar for #2.
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  3. #3
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    Re: Compound Angles

    Thanks slipeternal for replying i understand question 1 now but im still confused on the second part how would i implement that method in the second part.
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  4. #4
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    Re: Compound Angles

    Use the sum of angles formula to expand \sin(\theta+30^\circ):

    \begin{align*}10\sin \theta + 12\sin(\theta+30^\circ) & = 10 \sin \theta + 12(\sin \theta \cos 30^\circ + \cos \theta \sin 30^\circ) \\ & = (10+12\cos 30^\circ)\sin \theta + (12\sin 30^\circ)\cos \theta\end{align*}

    Evaluate \sin 30^\circ and \cos 30^\circ and then find a triangle whose sides have lengths 10+12\cos 30^\circ and 12\sin 30^\circ.
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