Moderate -0.3 Part (a) is a straightforward manipulation (divide by cos x) to find tan x = 3/7. Part (b) requires recognizing the structural similarity and substituting u = 2θ + 30°, then solving tan u = 3/7 and back-substituting. This is a standard 'hence' question testing pattern recognition and angle manipulation, slightly easier than average due to its routine nature and clear signposting.
8. (a) Given that \(7 \sin x = 3 \cos x\), find the exact value of \(\tan x\).
(b) Hence solve for \(0 \leqslant \theta < 360 ^ { \circ }\)
$$7 \sin \left( 2 \theta + 30 ^ { \circ } \right) = 3 \cos \left( 2 \theta + 30 ^ { \circ } \right)$$
giving your answers to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Exact equivalent accepted; recurring decimal \(0.428571...\) accepted but not rounded answer
Part (b):
Answer
Marks
Guidance
Answer/Working
Mark
Guidance
\(\tan(2\theta + 30) = \frac{3}{7}\)
B1ft
Correct equation or follow through from part (a); \(2\theta+30\) may be implied later by subtracting 30 then dividing by 2
\(\tan^{-1}\frac{3}{7}\) \((\alpha)\)
M1
Finds arctan of \(\frac{3}{7}\); implied by value e.g. 23.19 or just \(\tan^{-1}\frac{3}{7}\)
One of \(\theta =\) awrt 87 or awrt 177 or awrt 267 or awrt 357
A1
Follow through any of their final \(\theta\)s for \(\theta \pm 90n\) within range
A1ft
All of \(\theta = 86.6, \ 176.6, \ 266.6, \ 356.6\)
A1
All 4 correct to required accuracy; ignore extra answers outside range; lose last A mark for extra answers inside range
## Question 8:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\tan x = \frac{3}{7}$ | B1 | Exact equivalent accepted; recurring decimal $0.428571...$ accepted but not rounded answer |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\tan(2\theta + 30) = \frac{3}{7}$ | B1ft | Correct equation or follow through from part (a); $2\theta+30$ may be implied later by subtracting 30 then dividing by 2 |
| $\tan^{-1}\frac{3}{7}$ $(\alpha)$ | M1 | Finds arctan of $\frac{3}{7}$; implied by value e.g. 23.19 or just $\tan^{-1}\frac{3}{7}$ |
| One of $\theta =$ awrt 87 or awrt 177 or awrt 267 or awrt 357 | A1 | |
| Follow through any of their final $\theta$s for $\theta \pm 90n$ within range | A1ft | |
| All of $\theta = 86.6, \ 176.6, \ 266.6, \ 356.6$ | A1 | All 4 correct to required accuracy; ignore extra answers outside range; lose last A mark for extra answers inside range |
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