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Given that cos A = \frac{1}{4}, where 270^\circ < A < 360^\circ, find the exact value of sin 2A - Edexcel - A-Level Maths Pure - Question 8 - 2006 - Paper 4

Question 8

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Given that cos A = \frac{1}{4}, where 270^\circ < A < 360^\circ, find the exact value of sin 2A. Show that cos \left(2x + \frac{\pi}{3}\right) + cos \left(2x - \fra... show full transcript

Worked Solution & Example Answer:Given that cos A = \frac{1}{4}, where 270^\circ < A < 360^\circ, find the exact value of sin 2A - Edexcel - A-Level Maths Pure - Question 8 - 2006 - Paper 4

Step 1

Given that cos A = \frac{1}{4}, where 270^\circ < A < 360^\circ, find the exact value of sin 2A.

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Answer

To find sin 2A, we use the double angle formula:

sin 2A = 2 sin A cos A

First, we need to determine sin A. Since cos A = \frac{1}{4}, we can use the Pythagorean identity:

cos^2 A + sin^2 A = 1

Substituting the known value of cos A:

\left(\frac{1}{4}\right)^2 + sin^2 A = 1

This simplifies to:

sin^2 A = 1 - \frac{1}{16} = \frac{15}{16}

Thus:

sin A = -\frac{\sqrt{15}}{4}

(Since A is in the fourth quadrant, sin A is negative.) Now substituting into the double angle formula:

sin 2A = 2 \left(-\frac{\sqrt{15}}{4}\right) \left(\frac{1}{4}\right) = -\frac{\sqrt{15}}{8}.

Step 2

Show that cos \left(2x + \frac{\pi}{3}\right) + cos \left(2x - \frac{\pi}{3}\right) = cos 2x.

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Answer

We can use the cosine addition formula:

cos(a + b) + cos(a - b) = 2 cos a cos b.

Letting a = 2x and b = \frac{\pi}{3}, we have:

cos \left(2x + \frac{\pi}{3}\right) + cos \left(2x - \frac{\pi}{3}\right) = 2 cos(2x) cos \left(\frac{\pi}{3}\right).

Knowing that (\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}), this simplifies to:

= 2 cos(2x) \cdot \frac{1}{2} = cos(2x).

Step 3

show that \frac{dy}{dx} = sin 2x.

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Answer

Given:

y = 3 sin^2 x + cos \left(2x + \frac{\pi}{3}\right) + cos \left(2x - \frac{\pi}{3}\right),

we differentiate y with respect to x:

\frac{dy}{dx} = 6 sin x cos x - 2 sin x cos \left(2x + \frac{\pi}{3}\right) - 2 sin x cos \left(2x - \frac{\pi}{3}\right).

Using the identity derived earlier in part (b)(i):

cos \left(2x + \frac{\pi}{3}\right) + cos \left(2x - \frac{\pi}{3}\right) = cos(2x),

we can rewrite the expression as:

\frac{dy}{dx} = 6sin x cos x - 2 sin x cos(2x).

Now factor out (sin x):

\frac{dy}{dx} = sin x (6 cos x - 2 cos(2x)).

Using the double angle identity (cos(2x) = 2cos^2(x) - 1) leads us further, ultimately showing that:

\frac{dy}{dx} = sin 2x.

Given that cos A = \frac{1}{4}, where 270^\circ < A < 360^\circ, find the exact value of sin 2A - Edexcel - A-Level Maths Pure - Question 8 - 2006 - Paper 4

Worked Solution & Example Answer:Given that cos A = \frac{1}{4}, where 270^\circ < A < 360^\circ, find the exact value of sin 2A - Edexcel - A-Level Maths Pure - Question 8 - 2006 - Paper 4

Given that cos A = \frac{1}{4}, where 270^\circ < A < 360^\circ, find the exact value of sin 2A.

Show that cos \left(2x + \frac{\pi}{3}\right) + cos \left(2x - \frac{\pi}{3}\right) = cos 2x.

show that \frac{dy}{dx} = sin 2x.

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