Photo AI
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 $0 \leq x \leq 180^{\circ}$, $$\tan 2x = 5 \sin 2x$$ giving your answers to 1 decimal place where appropriate - Edexcel - A-Level Maths: Pure - Question 7 - 2012 - Paper 3
Question 7
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 $0 \leq x \leq 180^{\circ}$... 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 $0 \leq x \leq 180^{\circ}$, $$\tan 2x = 5 \sin 2x$$ giving your answers to 1 decimal place where appropriate - Edexcel - A-Level Maths: Pure - Question 7 - 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 show that the given equation can be rewritten in the required form, we start with the given equation:
By using the identity for tangent, we can write:
Thus, we can substitute this into our original equation:
Next, we multiply both sides by (\cos 2x) (assuming (\cos 2x \neq 0)):
Rearranging this gives:
Factoring out (\sin 2x) leads us to:
This confirms that the equation can indeed be written as:
Step 2
Hence solve, for 0 ≤ x ≤ 180°, tan 2x = 5 sin 2x
Answer
Now that we have established the equation
We can solve for when each factor is equal to zero:
-
First factor: (\sin 2x = 0)
- This implies:
- For , we find:
- When , which gives .
- When , which gives .
- When , , which gives .
Valid solutions are .
-
Second factor: (1 - 5 \cos 2x = 0)
- This implies:
- Therefore,
- Using a calculator:
- , so .
- And the solution in the range is:
- thus:
- .
Summarizing all solutions:
- .