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Given that cos A = \frac{\sqrt{3}}{4}, where 270° < A < 360°, find the exact value of sin 2A - Edexcel - A-Level Maths Pure - Question 1 - 2006 - Paper 4
Question 1
Given that cos A = \frac{\sqrt{3}}{4}, where 270° < A < 360°, find the exact value of sin 2A. (b) (i) Show that \cos \left(2x + \frac{\pi}{3}\right) + \cos \left(... show full transcript
Worked Solution & Example Answer:Given that cos A = \frac{\sqrt{3}}{4}, where 270° < A < 360°, find the exact value of sin 2A - Edexcel - A-Level Maths Pure - Question 1 - 2006 - Paper 4
Step 1
Given that cos A = \frac{\sqrt{3}}{4}, where 270° < A < 360°, find the exact value of sin 2A.
Answer
To find \sin 2A, we will use the double angle formula:
[ \sin 2A = 2 \sin A \cos A ]
First, we need to find \sin A. We know that \cos^2 A + \sin^2 A = 1.
Substituting \cos A:
[ \left(\frac{\sqrt{3}}{4}\right)^2 + \sin^2 A = 1 ]
[ \frac{3}{16} + \sin^2 A = 1 ]
[ \sin^2 A = 1 - \frac{3}{16} ]
[ \sin^2 A = \frac{16}{16} - \frac{3}{16} = \frac{13}{16} ]
Since 270° < A < 360°, \sin A is negative:
[ \sin A = -\sqrt{\frac{13}{16}} = -\frac{\sqrt{13}}{4} ]
Thus:
[ \sin 2A = 2 \times -\frac{\sqrt{13}}{4} \times \frac{\sqrt{3}}{4} = -\frac{\sqrt{39}}{8} ]
Step 2
Show that \cos \left(2x + \frac{\pi}{3}\right) + \cos \left(2x - \frac{\pi}{3}\right) = \cos 2x.
Answer
We can use the cosine addition formula:\n[\cos(A + B) + \cos(A - B) = 2 \cos A \cos B]
Let (A = 2x) and (B = \frac{\pi}{3}):
[\cos \left(2x + \frac{\pi}{3}\right) + \cos \left(2x - \frac{\pi}{3}\right) = 2 \cos(2x) \cos \left(\frac{\pi}{3}\right) ]
Since (\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}):
[= 2 \cos(2x) \cdot \frac{1}{2} = \cos(2x)]
Step 3
show that \frac{dy}{dx} = \sin 2x.
Answer
To differentiate (y = 3 \sin^3 x + \cos \left(2x + \frac{\pi}{3}\right) + \cos \left(2x - \frac{\pi}{3}\right)), apply the chain rule and product rule:
[\frac{dy}{dx} = 3 \cdot 3 \sin^2 x \cdot \cos x - \sin \left(2x + \frac{\pi}{3}\right) \cdot 2 - \sin \left(2x - \frac{\pi}{3}\right) \cdot 2]
Using the result from part (b)(i):
[= 2 \cos(2x) \cdot \frac{1}{2} = \sin(2x)]
Hence, (\frac{dy}{dx} = \sin 2x).