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8. (a) Show that the equation $$ \cos^2 x = 8 \sin^2 x - 6 \sin x $$ can be written in the form $$(3 \sin x - 1)^2 = 2$$ (b) Hence solve, for $0 \leq x < 360^\circ$, $$ \cos^2 x = 8 \sin^2 x - 6 \sin x $$ giving your answers to 2 decimal places. - Edexcel - A-Level Maths: Pure - Question 10 - 2017 - Paper 3
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8. (a) Show that the equation $$ \cos^2 x = 8 \sin^2 x - 6 \sin x $$ can be written in the form $$(3 \sin x - 1)^2 = 2$$ (b) Hence solve, for $0 \leq x < 360^\ci... show full transcript
Worked Solution & Example Answer:8. (a) Show that the equation $$ \cos^2 x = 8 \sin^2 x - 6 \sin x $$ can be written in the form $$(3 \sin x - 1)^2 = 2$$ (b) Hence solve, for $0 \leq x < 360^\circ$, $$ \cos^2 x = 8 \sin^2 x - 6 \sin x $$ giving your answers to 2 decimal places. - Edexcel - A-Level Maths: Pure - Question 10 - 2017 - Paper 3
Step 1
Show that the equation can be written in the form (3 sin x - 1)² = 2
Answer
To rewrite the equation, start from:
Using the Pythagorean identity, we can substitute for :
Rearranging gives:
This can be rearranged as:
Next, let . We can express the left-hand side in the form of a squared expression:
Thus, we have shown that
can indeed be written as the required form.
Step 2
Hence solve, for 0 ≤ x < 360°, cos²x = 8sin²x - 6sinx giving your answers to 2 decimal places.
Answer
Starting from the established form:
Taking the square root of both sides yields:
This leads to two cases:
Case 1:
Solving for x gives:
Case 2:
For this case, we find:
Thus, the final answers to 2 decimal places are: