A Level H1 Mathematics Geometry Trigonometry Quiz

A-Level Maths H1 Quiz - Geometry Trigonometry (Answer Key)

1.
$225 \times \frac{\pi}{180} = \frac{225\pi}{180} = \frac{5\pi}{4}$
Answer: $\frac{5\pi}{4}$
[1]

2.
Reference angle is $\frac{\pi}{6}$. Sine is negative in 3rd and 4th quadrants.
$\theta = \pi + \frac{\pi}{6} = \frac{7\pi}{6}$
$\theta = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}$
Answer: $\frac{7\pi}{6}, \frac{11\pi}{6}$
[2]

3.
Using $\sin^2 \alpha + \cos^2 \alpha = 1$:
$\sin \alpha = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{\frac{16}{25}} = \frac{4}{5}$
$\tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{4/5}{3/5} = \frac{4}{3}$
Answer: $\frac{4}{3}$
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4.
Numerator: $\sin^2 x + \cos^2 x = 1$.
Denominator: $\sec x = \frac{1}{\cos x}$.
$\frac{1}{\sec x} = \cos x$
Answer: $\cos x$
[1]

5.
$\frac{7\pi}{6}$ is in the 3rd quadrant. Reference angle is $\frac{\pi}{6}$.
Sine is negative in the 3rd quadrant.
$\sin\left(\frac{7\pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2}$
Answer: $-\frac{1}{2}$
[1]

6.
$2\cos^2 x = 1 \implies \cos^2 x = \frac{1}{2} \implies \cos x = \pm \frac{1}{\sqrt{2}}$
Reference angle is $\frac{\pi}{4}$.
Quadrants 1, 2, 3, 4 all have solutions.
$x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$
Answer: $\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$
[2]

7.
$\tan^2 \theta = \frac{1}{3} \implies \tan \theta = \pm \frac{1}{\sqrt{3}}$
Reference angle is $\frac{\pi}{6}$.
Range $-\pi \le \theta \le \pi$.
$\theta = \frac{\pi}{6}, -\frac{\pi}{6}, \pi - \frac{\pi}{6} = \frac{5\pi}{6}, -\pi + \frac{\pi}{6} = -\frac{5\pi}{6}$
Answer: $\pm \frac{\pi}{6}, \pm \frac{5\pi}{6}$
[2]

8.
LHS:
$\frac{1 - \cos^2 A}{\sin A}$
Using identity $\sin^2 A + \cos^2 A = 1 \implies 1 - \cos^2 A = \sin^2 A$:
$= \frac{\sin^2 A}{\sin A}$
$= \sin A$
$= \text{RHS}$
Answer: Shown
[2]

9.
Substitute $\sin^2 x = 1 - \cos^2 x$:
$2(1 - \cos^2 x) + 3\cos x = 0$
$2 - 2\cos^2 x + 3\cos x = 0$
$2\cos^2 x - 3\cos x - 2 = 0$
Factorise: $(2\cos x + 1)(\cos x - 2) = 0$.
$\cos x = 2$ (No solution, as $-1 \le \cos x \le 1$).
$\cos x = -\frac{1}{2}$.
Reference angle $\frac{\pi}{3}$. Cosine is negative in 2nd and 3rd quadrants.
$x = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \approx 2.09$
$x = \pi + \frac{\pi}{3} = \frac{4\pi}{3} \approx 4.19$
Answer: $2.09, 4.19$
[3]

10.
$\sin x = \frac{1}{3}$. $x$ is obtuse (2nd quadrant), so $\cos x$ is negative.
$\cos x = -\sqrt{1 - \sin^2 x} = -\sqrt{1 - \left(\frac{1}{3}\right)^2} = -\sqrt{\frac{8}{9}}$
$\cos x = -\frac{2\sqrt{2}}{3}$
Answer: $-\frac{2\sqrt{2}}{3}$
[2]

11.
Use identity $\cos(2\theta) = 1 - 2\sin^2 \theta$.
$1 - 2\sin^2 \theta = \sin \theta$
$2\sin^2 \theta + \sin \theta - 1 = 0$
Factorise: $(2\sin \theta - 1)(\sin \theta + 1) = 0$.
Case 1: $\sin \theta = \frac{1}{2}$.
$\theta = \frac{\pi}{6}, \frac{5\pi}{6}$
Case 2: $\sin \theta = -1$.
$\theta = \frac{3\pi}{2}$
Answer: $\frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}$
[3]

12.
$R = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
$\tan \alpha = \frac{4}{3} \implies \alpha = \arctan\left(\frac{4}{3}\right)$.
$\alpha \approx 0.927$ rad.
Answer: $5\sin(x + 0.927)$
[3]

13.
Cosine Rule: $a^2 = b^2 + c^2 - 2bc \cos A$.
$BC^2 = 6^2 + 8^2 - 2(6)(8)\cos(60^\circ)$
$BC^2 = 36 + 64 - 96(0.5)$
$BC^2 = 100 - 48 = 52$
$BC = \sqrt{52} = 2\sqrt{13} \approx 7.21 \text{ cm}$
Answer: $7.21$ cm
[2]

14.
Sine Rule: $\frac{\sin P}{QR} = \frac{\sin Q}{PR}$ is incorrect pairing. Correct: $\frac{\sin P}{QR} = \frac{\sin Q}{PR}$? No, sides are opposite angles.
$\frac{\sin P}{QR} = \frac{\sin Q}{PR}$? No.
$\frac{\sin P}{QR} = \frac{\sin R}{PQ}$? No.
Standard Sine Rule: $\frac{a}{\sin A} = \frac{b}{\sin B}$.
Here: $\frac{QR}{\sin P} = \frac{PQ}{\sin R}$? No, we want $\angle Q$ (at vertex Q, opposite side PR? No, side opposite Q is PR. We don't know PR).
Wait, we know $PQ=10$ (side $r$), $QR=7$ (side $p$), $\angle P = 30^\circ$.
We want $\angle Q$? No, Sine Rule finds angle opposite known side. We know side $QR=7$ (opposite $P$) and side $PQ=10$ (opposite $R$).
So we can find $\angle R$ first.
$\frac{7}{\sin 30^\circ} = \frac{10}{\sin R}$
$\sin R = \frac{10 \sin 30^\circ}{7} = \frac{5}{7} \approx 0.714$
$R_1 = \arcsin(5/7) \approx 45.6^\circ$
$R_2 = 180^\circ - 45.6^\circ = 134.4^\circ$
Check validity:
If $R = 45.6^\circ$, $Q = 180 - 30 - 45.6 = 104.4^\circ$. (Valid)
If $R = 134.4^\circ$, $Q = 180 - 30 - 134.4 = 15.6^\circ$. (Valid)
The question asks for $\angle PQR$ (which is angle $Q$).
Answer: $104.4^\circ, 15.6^\circ$
[3]

15.
Area of sector: $A = \frac{1}{2}r^2\theta = 20 \implies r^2\theta = 40 \implies \theta = \frac{40}{r^2}$.
Perimeter of sector: $P = 2r + r\theta = 18$.
Substitute $\theta$:
$2r + r\left(\frac{40}{r^2}\right) = 18$
$2r + \frac{40}{r} = 18$
Multiply by $r$:
$2r^2 + 40 = 18r$
$2r^2 - 18r + 40 = 0$
Divide by 2:
$r^2 - 9r + 20 = 0$
Answer: Shown
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16.
Factorise $r^2 - 9r + 20 = 0$:
$(r - 4)(r - 5) = 0$
$r = 4 \text{ or } r = 5$
Answer: $4, 5$
[2]

17.
$4 = 3 + 2\sin\left(\frac{\pi t}{6}\right)$
$1 = 2\sin\left(\frac{\pi t}{6}\right)$
$\sin\left(\frac{\pi t}{6}\right) = 0.5$
Let $u = \frac{\pi t}{6}$. $\sin u = 0.5$.
Principal value $u = \frac{\pi}{6}$.
Second value in range $0 \le t \le 12 \implies 0 \le u \le 2\pi$: $u = \frac{5\pi}{6}$.
Case 1: $\frac{\pi t}{6} = \frac{\pi}{6} \implies t = 1$.
Case 2: $\frac{\pi t}{6} = \frac{5\pi}{6} \implies t = 5$.
Answer: $1.00, 5.00$ hours
[3]

18.
Graph of $y = 2\cos x - 1$.
Amplitude 2, shifted down 1.
Max at $x=0$: $y = 2(1)-1 = 1$. Point $(0, 1)$.
Min at $x=\pi$: $y = 2(-1)-1 = -3$. Point $(\pi, -3)$.
End at $x=2\pi$: $y = 1$. Point $(2\pi, 1)$.
x-intercepts: $2\cos x - 1 = 0 \implies \cos x = 0.5$.
$x = \frac{\pi}{3}, \frac{5\pi}{3}$. Points $(\frac{\pi}{3}, 0), (\frac{5\pi}{3}, 0)$.
Answer: Sketch with labels $(0,1), (\pi,-3), (2\pi,1), (\frac{\pi}{3},0), (\frac{5\pi}{3},0)$.
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19.
Let $AB = h$. Let $DB = x$. Then $CB = x + 50$.
In $\triangle ABD$: $\tan 40^\circ = \frac{h}{x} \implies x = \frac{h}{\tan 40^\circ}$.
In $\triangle ABC$: $\tan 25^\circ = \frac{h}{x+50} \implies x+50 = \frac{h}{\tan 25^\circ}$.
Substitute $x$:
$\frac{h}{\tan 40^\circ} + 50 = \frac{h}{\tan 25^\circ}$
$50 = h\left(\frac{1}{\tan 25^\circ} - \frac{1}{\tan 40^\circ}\right)$
$h = \frac{50}{\cot 25^\circ - \cot 40^\circ}$
$h = \frac{50}{2.1445 - 1.1918} = \frac{50}{0.9527} \approx 52.48$
Answer: $52.5$ m
[3]

20.
$\sin x = \frac{1}{2}$ at $x = \frac{\pi}{6}$ and $x = \frac{5\pi}{6}$.
Sine is positive and greater than $0.5$ between these values (peak at $\frac{\pi}{2}$ is 1).
Answer: $\frac{\pi}{6} < x < \frac{5\pi}{6}$
[2]