Standard +0.8 This question requires knowledge of the cotangent double angle formula (cot 2x = (cot²x - 1)/(2cot x)), substitution to form a quadratic in cot x, solving the quadratic, then finding angles in the given range. It combines reciprocal trig functions with double angles and requires multiple non-trivial steps, making it moderately harder than average but still within standard A-level technique.
Solve a 3-term quadratic in \(\tan x\) and obtain a value of \(x\)
DM1
Obtain answer e.g. \(x=35.1°\)
A1
Obtain second answer e.g. \(x=99.9°\), and no other in \((0°,180°)\)
A1
Ignore answers outside \((0°,180°)\). Treat answers in radians \((0.612,1.74)\) as a misread
Total: 6 marks
## Question 3:
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct trigonometric formulae to form an equation in $\tan x$ | *M1 | e.g. $\frac{1-\tan^2 x}{\tan x}+\frac{3}{\tan x}=5$ |
| Obtain a correct linear equation in any form | A1 | $1-\tan^2 x+3=5\tan x$ |
| Reduce equation to a 3-term quadratic | A1 | $\tan^2 x+5\tan x-4=0$, or 3-term equivalent |
| Solve a 3-term quadratic in $\tan x$ and obtain a value of $x$ | DM1 | |
| Obtain answer e.g. $x=35.1°$ | A1 | |
| Obtain second answer e.g. $x=99.9°$, and no other in $(0°, 180°)$ | A1 | Ignore answers outside $(0°,180°)$. Treat answers in radians $(0.612, 1.74)$ as a misread |
**Alternative method:**
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct formulae for $\sin 2x$ and $\cos 2x$ to form equation in $\sin x$ and $\cos x$ | *M1 | |
| Obtain $4\frac{\cos x}{\sin x}-\frac{\sin x}{\cos x}=5$ | A1 | |
| Reduce equation to a 3-term quadratic | A1 | $\tan^2 x+5\tan x-4=0$, or 3-term equivalent |
| Solve a 3-term quadratic in $\tan x$ and obtain a value of $x$ | DM1 | |
| Obtain answer e.g. $x=35.1°$ | A1 | |
| Obtain second answer e.g. $x=99.9°$, and no other in $(0°,180°)$ | A1 | Ignore answers outside $(0°,180°)$. Treat answers in radians $(0.612,1.74)$ as a misread |
**Total: 6 marks**
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