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6. (a) Show that the equation tan 2x = 5 sin 2x can be written in the form (1 - 5 cos 2x) sin 2x = 0 (b) Hence solve, for 0 ≤ x ≤ 180°, tan 2x = 5 sin 2x giving your answers to 1 decimal place where appropriate - Edexcel - A-Level Maths Pure - Question 8 - 2012 - Paper 3
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6. (a) Show that the equation tan 2x = 5 sin 2x can be written in the form (1 - 5 cos 2x) sin 2x = 0 (b) Hence solve, for 0 ≤ x ≤ 180°, tan 2x = 5 sin 2x giving ... show full transcript
Worked Solution & Example Answer:6. (a) Show that the equation tan 2x = 5 sin 2x can be written in the form (1 - 5 cos 2x) sin 2x = 0 (b) Hence solve, for 0 ≤ x ≤ 180°, tan 2x = 5 sin 2x giving your answers to 1 decimal place where appropriate - Edexcel - A-Level Maths Pure - Question 8 - 2012 - Paper 3
Step 1
Show that the equation tan 2x = 5 sin 2x can be written in the form (1 - 5 cos 2x) sin 2x = 0
Answer
To start, we use the identity for tangent, which states that . Thus we can rewrite the equation:
Next, multiply both sides by (\cos 2x) (assuming (\cos 2x \neq 0)):
Now rearranging, we can subtract (5\sin 2x\cos 2x) from both sides:
Factor out (\sin 2x):
Thus, we have shown that the equation can be written as:
.
Step 2
Hence solve, for 0 ≤ x ≤ 180°, tan 2x = 5 sin 2x
Answer
From the factored equation, we have two cases to consider:
- (\sin 2x = 0)
- (1 - 5 \cos 2x = 0)
Case 1: (\sin 2x = 0)
This gives us:
(2x = 0, 180, 360, ...)
Thus, performing the division:
(x = 0, 90, 180)
Case 2: (1 - 5 \cos 2x = 0)
Rearranging gives:
(5 \cos 2x = 1)
Thus, we find:
(\cos 2x = \frac{1}{5})
Using the inverse cosine function:
(2x = \cos^{-1}\left(\frac{1}{5}\right)) and (2x = 360 - \cos^{-1}\left(\frac{1}{5}\right))
Calculating the angles:
(2x \approx 78.46^\circ) and (2x \approx 281.54^\circ)
Dividing by 2:
(x \approx 39.23^\circ) (valid) and (x \approx 140.77^\circ) (valid)
Thus, combining both cases, the solutions are:
(x = 0, 39.2, 90, 140.8, 180)
Rounded to one decimal place, the answers are:
(x \approx 39.2^\circ, 140.8^\circ).