C3 Maths Question

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    Hi, bit stuck on this question:

    4 + root2(cos2x) + root2(sin2x)

    Find the greatest and least values of these expressions and the values of x for which they occur.

    How would I go about this?

    Thanks a lot
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    (Original post by rampallian)
    Hi, bit stuck on this question:

    4 + root2(cos2x) + root2(sin2x)

    Find the greatest and least values of these expressions and the values of x for which they occur.

    How would I go about this?

    Thanks a lot
    The problem is that there is a cos2x and a sin2x there as two terms.. maybe they could be combined into one, which would simplify the problem. How can you do that?
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    (Original post by rampallian)
    Hi, bit stuck on this question:

    4 + root2(cos2x) + root2(sin2x)

    Find the greatest and least values of these expressions and the values of x for which they occur.

    How would I go about this?

    Thanks a lot
    Few ways of doing this. One is to differentiate and set that to 0.
    Other way is to notice that  \sqrt 2 \cos 2x + \sqrt 2 \sin 2x \equiv 2[\frac{\sqrt 2}{2} \cos 2x +\frac{\sqrt 2}{2} \sin 2x] and use the addition formula for cosine or sine.
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    (Original post by rampallian)
    Hi, bit stuck on this question:

    4 + root2(cos2x) + root2(sin2x)

    Find the greatest and least values of these expressions and the values of x for which they occur.

    How would I go about this?

    Thanks a lot
    Firstly put it in the form 4+\sqrt{2}(\sin{2x}+\cos{2x}) and you can turn the bracket into R\sin(2x+\alpha) form. Then it should become easier from this point onward.

    Another way to do it is to say that y=4+\sqrt{2}(\sin2x + \cos2x)

    so \frac{dy}{dx}=\sqrt{2}(2\cos{2x} - 2\sin{2x}) and set this to 0.

    We know that y is a trigonometric function summing up sine and cosine therefore it is a wave-y graph, therefore maximum and minimum values are when the gradient is 0.

    So 0=2\sqrt{2}(\cos2x - \sin2x) and you can get \tan{2x} out of it and solve for x. Plug your values into the second differential to determine whether they are maxima or minima, and hence find these values using the original function.
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    Thanks, worked out how to do this now
 
 
 
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