June 2023 Paper 1 Q10
10 The curve with equation
\[y = \sin x^\circ\]for \(-360 \leqslant x \leqslant 360\) is shown below.
(a) Point \(A\) on the curve has coordinates \((a,\ 0.5)\)
(i) Find the value of \(a\) [2 marks]
(ii) State the value of \(\sin(180^\circ - a^\circ)\) [1 mark]
(b) Point \(B\) on the curve has coordinates \(\left(b,\ -\dfrac{3}{7}\right)\)
(i) Find the exact value of \(\sin(b^\circ - 180^\circ)\) [2 marks]
(ii) Find the exact value of \(\cos b^\circ\) [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| (i) States or uses sin 30 = 0.5 PI by sight of \(\pm 150\) or \(\pm 30\) or \(-330\) Maybe seen on diagram | M1 | 1.1a |
| Obtains -210 | A1 | 1.1b |
| (2) | ||
| (ii) Obtains \(0.5\) | B1 | 1.1b |
| (1) |
Typical solution
(i)
–180-30 = -210
(ii)
\[0.5\]| Scheme | Marks | AO |
|---|---|---|
| (i) Uses a correct approach to find \(\sin(b - 180)\). Might see \(\sin(205.37\ldots \pm 180)\) PI by correct answer or \(\sin(\pm 180 - 25.376\ldots)\) PI by correct answer or Correct use of compound angle formula PI by correct answer | M1 | 3.1a |
| Deduces \(\sin(b - 180) = \dfrac{3}{7}\) CAO | R1 | 2.2a |
| (2) | ||
| (ii) Uses \(\cos^2 x + \sin^2 x = 1\) or Draws right angled triangle with 3 and 7 on opp and hyp sides. PI by \(\cos b = -\dfrac{2\sqrt{10}}{7}\) OE exact form | M1 | 3.1a |
| Obtains \(\cos^2 b = \dfrac{40}{49}\) Condone \(b\) replaced by different variable or obtains a ratio for cosine of the correct exact magnitude. | A1 | 1.1b |
| Deduces \(\cos b = -\dfrac{2\sqrt{10}}{7}\) OE exact form CAO | R1 | 2.2a |
| (3) | ||
| (8 marks) |
Typical solution
(i)
\[\sin(b - 180) = -\sin b\]\[= \frac{3}{7}\](ii)
\[\cos^2 b + \left(-\frac{3}{7}\right)^2 = 1\]\[\cos^2 b = \frac{40}{49}\]\[\cos b = -\frac{2\sqrt{10}}{7}\]