CAIE IGCSE Additional Math 10 Trigonometry Topic Practice

$\frac{2}{\sqrt{3}}$ or $\frac{2\sqrt{3}}{3}$
B2 for $[BM=]\frac{\sqrt{3}a}{2}$ or $\sec^2 30=1+\left(\frac{1}{\sqrt{3}}\right)^2$ or $\cos^2 30=1-\left(\frac{a/2}{a}\right)^2$ oe
B1 for $[BM^2=]a^2-\frac{a^2}{4}$ soi

$\frac{2\sec x}{(\sec x-1)(\sec x+1)}$ soi
B1 for adding the two fractions.
$\frac{2\sec x}{\sec^2 x-1}$
B1 dep on previous B1 for expanding and simplifying the denominator.
$\frac{2\sec x}{\tan^2 x}$ or $\frac{2}{\tan^2 x\cos x}$
B1 dep on previous B1 for use of $\sec^2 x=1+\tan^2 x$.
$\frac{2\sec x}{\tan x}\times\cot x$ or $\frac{2\cos x}{\sin^2 x}$ oe or $\frac{2}{\tan x\sin x}$, leading to $2\operatorname{cosec}x\cot x$ AG
B1 dep on previous B1 with at least one more correct step to get to the given answer.
Alternative method:
$$\begin{aligned} \frac{1}{\frac{1}{\cos x}-1}+\frac{1}{\frac{1}{\cos x}+1}&=\frac{\cos x}{1-\cos x}+\frac{\cos x}{1+\cos x} \end{aligned}$$$=\frac{\cos x+\cos^2 x+\cos x-\cos^2 x}{1-\cos^2 x}$
(B1) dep on previous B1 for adding the two fractions.
$=\frac{2\cos x}{\sin^2 x}$
(B1) dep on previous B1 for use of $\sin^2 x=1-\cos^2 x$.
$=2\operatorname{cosec}x\times\frac{\cos x}{\sin x}=2\operatorname{cosec}x\cot x$ AG

$180^{\circ}$ or $\pi$
Allow without degrees sign

Correct curve

B1 for amplitude 3

B1 for period $180^{\circ}$

B1 for correct end points

$\frac{\sin \theta}{\sqrt{\cot ^{2} \theta}}+\frac{1}{\sqrt{\sec ^{2} \theta}}$
for correct use of the trigonometry
identity
$\operatorname{cosec}^{2} \theta-1=\cot ^{2} \theta$$\sin \theta \tan \theta+\cos \theta$ oe
$\sin \theta \times \frac{\sin \theta}{\cos \theta}+\cos \theta$$\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\cos \theta}=\frac{1}{\cos \theta}=\sec \theta$
Dep on all 3 previous marks
Last mark is not available if persistent
missing of $\theta$

Alternative

$\frac{\sin \theta}{\sqrt{\cot ^{2} \theta}}+\frac{1}{\sqrt{\sec ^{2} \theta}}$
for correct use of the trigonometry
identity
$\operatorname{cosec}^{2} \theta-1=\cot ^{2} \theta$$\frac{\sec \theta \sin \theta+\cot \theta}{\cot \theta \sec \theta}$$\frac{\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}}{\frac{\cos \theta}{\sin \theta} \times \frac{1}{\cos \theta}}$$$\frac{\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\cos \theta \sin \theta}}{\frac{1}{\sin \theta}} \text { leading to } \frac{1}{\cos \theta}$$
Dep on all 3 previous marks
Last mark is not available if persistent
missing $\theta$

$\cos x=\frac{1}{\alpha}$
could be implied by $\frac{1}{\alpha^{2}}=1-\sin ^{2} x$$\sin ^{2} x+\left(\frac{1}{\alpha}\right)^{2}=1$ oe

OR [opposite ${ }^{2}=$ ] $\alpha^{2}-1$ soi and use of $\sin x=\frac{\text { opposite }}{\text { hypotenuse }}$ at some stage leading to $\sin x=[ \pm] \sqrt{1-\frac{1}{\alpha^{2}}}$ oe
$\sin x=-\sqrt{1-\frac{1}{\alpha^{2}}}$ oe

$[\tan2x=0]$ any two of 2x=0,180,360x=0,90,180 nfww and no extras from $\tan2x=0$$[\tan2x=4]$$2x=75.96\ldots$ or $255.96\ldots$x=38.0 or 37.9818..., and x=128.0 or 127.9818..., with no extras

$\sin(y+1.2)=\frac14$
Two correct values for y+1.2 soi:
$$y+1.2=0.252680\ldots,\quad y+1.2=\pi-0.252680\ldots=2.88891\ldots,\quad y+1.2=-\pi-0.252680\ldots=-3.39427\ldots$$
M1 for any one correct value.
$y=-0.947\ldots$, $1.68891\ldots$, $-4.59427\ldots$, and no others in -5<y<2