Sub-topic 1
2025 · National One-Eighth
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$\cos(a-b)$ from $\sin a$, $\cos b$ at standard angles $30^\circ$, $45^\circ$, $60^\circ$
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Sub-topic 2
2024 · National Preliminary
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$\cos2a$ given $\cos a=-\dfrac{5}{13}$
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Sub-topic 3
2025 · National One-Eighth
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Continuity of $\tan x$, $\sin x$, $\cos x$ and other ratios (true/false)
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Sub-topic 4
2025 · National One-Eighth
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Finding trig ratios via pythagorean identities
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Sub-topic 5
2025 · National One-Eighth
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Parametric identity $(x+1)^2+(y-3)^2=4$ (true/false)
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Sub-topic 6
2025 · National One-Eighth
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Trig ratios table and key identities: applying the reference identities
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Sub-topic 7
2024 · National Preliminary
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Rationalising $\dfrac{a}{1\pm\sin\theta\text{ or }\cos\theta}$ at standard angles
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Sub-topic 8
2026 · National One-Eighth
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Riddles: pythagorean and compound-angle identities
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Sub-topic 9
2026 · National Preliminary
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Evaluating $\cos15^\circ$, $\sin15^\circ$, $\cos75^\circ$ etc via the $45^\circ$-$30^\circ$ decomposition
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Sub-topic 10
2026 · National Preliminary
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Evaluating $\cos\dfrac{\pi}{12}$ via $\dfrac{\pi}{3}-\dfrac{\pi}{4}$
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Sub-topic 11
2026 · National Preliminary
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Eliminating $\theta$ from $x=2\cos\theta$, $y=3\sin\theta$
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Sub-topic 12
2025 · National Preliminary
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$\sin(a+b)=1$ from pythagorean-triple ratios
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Sub-topic 13
2026 · National Semifinal
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Expressing a tangent expression as $\tan(\theta+\alpha)$
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Sub-topic 14
2026 · National Semifinal
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$\sin\theta\sin\phi$ from $\cos(\theta+\phi)$ and $\cos(\theta-\phi)$
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Sub-topic 15
2026 · National Quarterfinal
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$\tan x$ from $\sin\left(x+\dfrac{\pi}{4}\right)=\cos x$
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Sub-topic 16
2026 · National Quarterfinal
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Evaluating $\sin\dfrac{7\pi}{12}$ via $\dfrac{\pi}{4}+\dfrac{\pi}{3}$
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Sub-topic 17
2026 · National Quarterfinal
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Expressing a tangent quotient as $\tan(\theta+\alpha)$ or $\tan(\alpha-\theta)$
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Sub-topic 18
2026 · National Quarterfinal
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Evaluating $\sin^2\dfrac{\pi}{12}$ via the half-angle identity
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Sub-topic 19
2026 · National One-Eighth
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$\tan x$ from $\sin\left(x+\dfrac{\pi}{4}\right)=\cos x$
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Sub-topic 20
2021 · National Preliminary
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Signs of trig ratios for an obtuse angle (true/false)
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Sub-topic 21
2026 · National Preliminary
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Related-angle equalities such as $\sin137^\circ=\sin43^\circ$ (true/false)
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Sub-topic 22
2026 · National Preliminary
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Related-angle equalities (true/false)
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Sub-topic 23
2026 · National Preliminary
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Related-angle equalities (true/false)
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Sub-topic 24
2024 · National One-Eighth
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Related-angle equalities for $(2\pi-x)$ (true/false)
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Sub-topic 25
2026 · National Final
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Simplifying $(\cos\theta+\sin\theta)(\cos\theta-\sin\theta)$
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Sub-topic 26
2026 · National One-Eighth
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$\tan x$ from $\sin\left(x+\dfrac{\pi}{4}\right)=\cos x$
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Sub-topic 27
2024 · Regional Championship
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Evaluating $\dfrac{\tan45^\circ-\tan60^\circ}{\tan45^\circ+\tan60^\circ}$
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Sub-topic 28
2026 · National Preliminary
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Riddle: cotangent, the reciprocal of tangent
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Sub-topic 29
2025 · National Preliminary
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Evaluating $\sin(2\theta)$ given $\sin\theta\cos\theta=\dfrac{2}{5}$
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Sub-topic 30
2025 · National Preliminary
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$\sin(a+b)=1$ from pythagorean-triple ratios, $\sin a=\cos b=\dfrac{3}{5}$
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Sub-topic 31
2024 · National Preliminary
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Evaluating $\sin25^\circ\sin20^\circ-\cos25^\circ\cos20^\circ$ and similar compound-angle expressions
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Sub-topic 32
2013 · National Preliminary
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Riddles: pythagoras' theorem, the cosine rule and the sine rule
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Sub-topic 33
2013 · National Preliminary
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The double-angle identity for sine is not $2\sin a$ (true/false)
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Sub-topic 34
2013 · National Preliminary
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The double-angle identity for cosine as $1-2\sin^2a$ (true/false)
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Sub-topic 35
2013 · National Preliminary
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The cosine difference formula (true/false)
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Sub-topic 36
2013 · National Quarterfinal
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Expressing double-angle and addition-formula expressions as a single ratio
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Sub-topic 37
2013 · National Quarterfinal
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Evaluating $\cos77^\circ\cos17^\circ+\sin77^\circ\sin17^\circ$ and $2\sin150^\circ$
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Sub-topic 38
2014 · National Quarterfinal
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Co-function identity $\cos(90^\circ-a)=\sin a$ (true/false)
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Sub-topic 39
2014 · National Quarterfinal
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Supplementary-angle identity $\sin(180^\circ-a)=\sin a$ (true/false)
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Sub-topic 40
2014 · National Quarterfinal
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Supplementary-angle identity $\tan(180^\circ-a)=-\tan a$ (true/false)
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Sub-topic 41
2016 · National One-Eighth
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Co-function identity $\cos(90^\circ-a)=\sin a$ (true/false)
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Sub-topic 42
2016 · National One-Eighth
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Co-function identity $\cos(90^\circ+a)=-\sin a$ (true/false)
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Sub-topic 43
2016 · National One-Eighth
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Co-function identity $\sin(90^\circ+a)=\cos a$ (true/false)
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Sub-topic 44
2016 · National Preliminary
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Double-angle tangent identity from a given $\tan a$
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Sub-topic 45
2018 · National Preliminary
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Evaluating $2\cos^2(15^\circ)-1$
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Sub-topic 46
2018 · National Preliminary
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Does $\sin x$ increase steadily on $0\lt x\lt\pi$? (true/false)
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Sub-topic 47
2018 · National Preliminary
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Does $\sin x$ decrease steadily on $0\lt x\lt\pi$? (true/false)
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Sub-topic 48
2018 · National Preliminary
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Does $\sin x$ both increase and decrease on $0\lt x\lt\pi$? (true/false)
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Sub-topic 49
2019 · National Preliminary
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Evaluating $\dfrac{\tan87^\circ-\tan42^\circ}{1+\tan87^\circ\tan42^\circ}$
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Sub-topic 50
2021 · National Quarterfinal
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$\sin(2\alpha)$ from $\sin\alpha=\dfrac{1}{\sqrt{10}}$, $\alpha$ acute
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Sub-topic 51
2021 · National Quarterfinal
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$\tan(2\alpha)$ from $\sin\alpha=\dfrac{1}{\sqrt{10}}$, $\alpha$ acute
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Sub-topic 52
2021 · National Quarterfinal
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$\cos(2\alpha)$ from $\sin\alpha=\dfrac{1}{\sqrt{10}}$, $\alpha$ acute
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Sub-topic 53
2022 · National Preliminary
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Evaluating $\tan(45^\circ+\theta)$ given $\tan\theta=2$
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Sub-topic 54
2022 · National Preliminary
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Evaluating $\tan(45^\circ-\theta)$ given $\tan\theta=2$
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Sub-topic 55
2022 · National Preliminary
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Evaluating $\tan(30^\circ+\theta)$ given $\tan\theta=2$
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Sub-topic 56
2022 · National One-Eighth
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Is $\sin137^\circ=\sin43^\circ$? (true/false)
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Sub-topic 57
2022 · National One-Eighth
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Is $\cos125^\circ=\cos55^\circ$? (true/false)
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Sub-topic 58
2022 · National One-Eighth
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Is $\tan113^\circ=\tan67^\circ$? (true/false)
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Sub-topic 59
2023 · National Preliminary
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Finding $\sin x$ given $\cos x=-\dfrac{2}{5}$
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Sub-topic 60
2023 · National Preliminary
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Finding $\cos x$ given $\sin x=\dfrac{3}{4}$
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Sub-topic 61
2023 · National Preliminary
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Finding $\sec x$ given $\tan x=2$
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Sub-topic 62
2023 · National Preliminary
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$\sin(2a)$ given $\sin a=\dfrac{3}{5}$, $a$ obtuse
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Sub-topic 63
2023 · National Preliminary
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Is $\tan^2a=\sec^2a+1$? (true/false)
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Sub-topic 64
2023 · National Preliminary
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Is $\csc^2a=\cot^2a+1$? (true/false)
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Sub-topic 65
2023 · National Preliminary
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Is $\cos^2a-\sin^2a=\cos(2a)$? (true/false)
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Sub-topic 66
2023 · National One-Eighth
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Is $\sin57^\circ\cos63^\circ+\cos57^\circ\sin63^\circ=\dfrac{\sqrt3}{2}$? (true/false)
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Sub-topic 67
2023 · National One-Eighth
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Is $\cos(47^\circ+73^\circ)-\sin47^\circ\sin73^\circ=\dfrac12$? (true/false)
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Sub-topic 68
2023 · National One-Eighth
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Is $\dfrac{\tan59^\circ+\tan61^\circ}{1-\tan59^\circ\tan61^\circ}=\sqrt3$? (true/false)
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Sub-topic 69
2023 · National Preliminary
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Exact value of $\tan\dfrac{7\pi}{12}$
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Sub-topic 70
2023 · National Preliminary
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Exact value of $\sin\dfrac{7\pi}{12}$
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Sub-topic 71
2023 · National Preliminary
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Exact value of $\cos\dfrac{7\pi}{12}$
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Sub-topic 72
2023 · National Preliminary
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Finding $\sin x$ given $\cos x=\dfrac{5}{8}$
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