Standard +0.3 This question requires knowing the double angle formula (cos 2θ = 1 - 2sin²θ) and that cosec θ = 1/sin θ, then solving a quadratic in sin θ. It's a standard C3 exercise with clear signposting ('by first using appropriate identities') and straightforward algebraic manipulation, making it slightly easier than average.
State or imply \(\cosec\theta = 1 \div \sin\theta\)
B1
allow \(\cosec = 1 \div \sin\)
Attempt to express equation in terms of \(\sin\theta\) only
M1
using identity of form \(\pm 1 \pm 2\sin^2\theta\) for \(\cos 2\theta\)
Obtain \(10\sin^2\theta + 2\sin\theta - 5 = 0\)
A1
or unsimplified equiv involving \(\sin\theta\) only but with no \(\sin\theta\) remaining in denominator
Attempt use of formula to find \(\sin\theta\) from 3-term quadratic equation involving \(\sin\theta\) (using formula or completing square even if their equation can be solved by factorisation)
M1
use implied by at least one correct value of \(\sin\theta\) or \(\theta\); if correct quadratic formula quoted, condone one sign error for M1; if formula not first quoted, any error leads to M0. If completion of square used to solve equation, this must be correct for M1 to be earned
Obtain \(37.9°\)
A1
or greater accuracy \(37.8896\ldots\)
Obtain \(142°\)
A1
or greater accuracy \(142.1103\ldots\); and no others between 0 and 180; ignore any answers outside \(0\)–\(180\). No working and answers only (max 2/6): 37.9 (or greater accuracy) B1; 142 (or greater accuracy) and no others … B1
Total
[6]
## Question 2:
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $\cosec\theta = 1 \div \sin\theta$ | B1 | allow $\cosec = 1 \div \sin$ |
| Attempt to express equation in terms of $\sin\theta$ only | M1 | using identity of form $\pm 1 \pm 2\sin^2\theta$ for $\cos 2\theta$ |
| Obtain $10\sin^2\theta + 2\sin\theta - 5 = 0$ | A1 | or unsimplified equiv involving $\sin\theta$ only but with no $\sin\theta$ remaining in denominator |
| Attempt use of formula to find $\sin\theta$ from 3-term quadratic equation involving $\sin\theta$ (using formula or completing square even if their equation can be solved by factorisation) | M1 | use implied by at least one correct value of $\sin\theta$ or $\theta$; if correct quadratic formula quoted, condone one sign error for M1; if formula not first quoted, any error leads to M0. If completion of square used to solve equation, this must be correct for M1 to be earned |
| Obtain $37.9°$ | A1 | or greater accuracy $37.8896\ldots$ |
| Obtain $142°$ | A1 | or greater accuracy $142.1103\ldots$; and no others between 0 and 180; ignore any answers outside $0$–$180$. No working and answers only (max 2/6): 37.9 (or greater accuracy) B1; 142 (or greater accuracy) and no others … B1 |
| **Total** | **[6]** | |
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