A-Level Edexcel Maths Pure Integration: In this question you must show a

In this question you must show all stages of your working - Edexcel - A-Level Maths Pure - Question 12 - 2020 - Paper 2

Question 12

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. (a) Show that $$\cos 3A \equiv 4... show full transcript

Worked Solution & Example Answer:In this question you must show all stages of your working - Edexcel - A-Level Maths Pure - Question 12 - 2020 - Paper 2

Step 1

Show that $ \cos 3A \equiv 4\cos^3 A - 3\cos A $

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Answer

To prove the identity, we can use the angle subtraction formula for cosine. We know that:

$\cos 3A = \cos(2A + A) = \cos 2A \cos A - \sin 2A \sin A$

Next, we apply the double angle identities:

( \cos 2A = 2\cos^2 A - 1 )
( \sin 2A = 2\sin A \cos A )

Thus, substituting the double angle formulas:

$\cos 3A = (2\cos^2 A - 1)\cos A - (2\sin A \cos A)\sin A$

This gives us:

$\cos 3A = 2\cos^3 A - \cos A - 2\sin^2 A \cos A$

Since ( \sin^2 A = 1 - \cos^2 A ), we can substitute:

$\cos 3A = 2\cos^3 A - \cos A - 2(1 - \cos^2 A)\cos A$

Simplifying this yields:

$\cos 3A = 4\cos^3 A - 3\cos A$

Step 2

Hence solve, for $-90^\circ \leq r \leq 180^\circ$, the equation $ 1 - \cos 3x = \sin x $

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Answer

Using the result from part (a):

Substitute ( \cos 3x ):

$1 - (4\cos^3 x - 3\cos x) = \sin x$

This simplifies to:

$3\cos x - 4\cos^3 x = \sin x$
Rearrange this into a cubic equation:

$4\cos^3 x - 3\cos x + \sin x = 0$
Next, we use the identity ( \sin x = \sqrt{1 - \cos^2 x} ) to convert this into a polynomial in terms of ( y = \cos x ):
We will check the possible values for x:
- When ( x = 0: \ 4(1)^3 - 3(1) + 0 = 1\ \Rightarrow \text{not a solution})
- When ( x = 60^\circ: \ 4(\frac{1}{2})^3 - 3(\frac{1}{2}) + \sqrt{1 - (\frac{1}{2})^2} = 1 \Rightarrow \text{valid solution})
Continue checking within the interval for other possible solutions.
- Consider negative angles too, such as -90°, for a full set of solutions within the specified range.

In this question you must show all stages of your working - Edexcel - A-Level Maths Pure - Question 12 - 2020 - Paper 2

Worked Solution & Example Answer:In this question you must show all stages of your working - Edexcel - A-Level Maths Pure - Question 12 - 2020 - Paper 2

Show that \( \cos 3A \equiv 4\cos^3 A - 3\cos A \)

Hence solve, for $-90^\circ \leq r \leq 180^\circ$, the equation \( 1 - \cos 3x = \sin x \)

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