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1.5.14 Harmonic form Rcos / Rsin (A-level only)

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Question 1

Let f(x)=6sin⁡xcos⁡x+6cos⁡2x−1f(x) = 6 \sin x \cos x + 6 \cos^2 x - 1f(x)=6sinxcosx+6cos2x−1.

a.

Write f(x)f(x)f(x) in the form

asin⁡2x+bcos⁡2x+c a \sin 2x + b \cos 2x + c asin2x+bcos2x+c

where a,b,a, b,a,b, and ccc are integers to be found.

[3]
b.

Use the answer to part (a) to write f(x)f(x)f(x) in the form

Rsin⁡(2x+α)+c R \sin (2x + \alpha) + c Rsin(2x+α)+c

where R>0R > 0R>0 and 0<α<π20 < \alpha < \frac{\pi}{2}0<α<2π​. Give the exact value of RRR and give the value of α\alphaα in radians to 3 significant figures.

[3]
c.

Hence, or otherwise, (i) state the maximum value of f(x)f(x)f(x), (ii) find the second smallest positive value of xxx at which a maximum value of f(x)f(x)f(x) occurs. Give your answer to 3 significant figures.

[3]

1.5.14 Harmonic form Rcos / Rsin (A-level only) Questions

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