Moderate -0.8 This is a straightforward application of the Pythagorean identity sin²θ + cos²θ = 1 with a simple decimal value. The quadrant information tells students cos θ is negative. Requires only one identity and basic arithmetic, making it easier than average.
ACF; or uses \(\sin\theta = -0.1\) and right-angled triangle to get magnitude of \(\cos\theta\); or obtains \(\cos^2\theta = 0.99\) CAO
\(\cos\theta = -\dfrac{3}{10}\sqrt{11}\)
A1
Accept \(\cos\theta = -\sqrt{0.99}\) or exact equivalent \(-\dfrac{3}{10}\sqrt{11}\); solves and selects correct sign; ISW if exact answer seen and then evaluated; NB any full numerical approach scores M0A0
Total: 2 marks
## Question 3:
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitutes $\sin\theta = -0.1$ into $\sin^2\theta + \cos^2\theta = 1$, giving $0.01 + \cos^2\theta = 1$, obtaining $\cos^2\theta = 0.99$ | M1 | ACF; or uses $\sin\theta = -0.1$ and right-angled triangle to get magnitude of $\cos\theta$; or obtains $\cos^2\theta = 0.99$ CAO |
| $\cos\theta = -\dfrac{3}{10}\sqrt{11}$ | A1 | Accept $\cos\theta = -\sqrt{0.99}$ or exact equivalent $-\dfrac{3}{10}\sqrt{11}$; solves and selects correct sign; ISW if exact answer seen and then evaluated; NB any full numerical approach scores M0A0 |
**Total: 2 marks**
3 It is given that $\sin \theta = - 0.1$ and $180 ^ { \circ } < \theta < 270 ^ { \circ }$
Find the exact value of $\cos \theta$
\hfill \mbox{\textit{AQA AS Paper 2 2019 Q3 [2]}}