Progressive Trigonometry practice questions
Use Clevolab for progressive practice in trigonometry. This page shows what the topic covers, what skills the current set targets, and a few real examples from the reviewed question bank.
About this topic
Clevolab treats trigonometry as repeated practice with explanation, not just answer checking. This page is designed to make the topic legible before you open the app.
The current Progressive set covers triangle problems, angle reasoning, trigonometric ratios, and standard trigonometric methods. 143 reviewed questions currently published for this page.
What you can practise
- Sine, cosine, and tangent
- Right-angled triangle methods
- Sine rule and cosine rule
- Angle problems
- Practical geometry links
Real sample questions from the current set
These examples come from the reviewed questions currently stored for this topic. They are here so the page shows the actual flavour of Clevolab, not just a summary.
Sample question
Which ratio is $\tan\theta$?
- Ahypotenuse over adjacentAnswer option
- Bopposite over adjacentCorrect answer
- Cadjacent over hypotenuseAnswer option
- Dhypotenuse over oppositeAnswer option
Why this answer is right
Tangent measures the steepness as opposite over adjacent. This comes from SOH CAH TOA, where TOA means tangent equals opposite over adjacent.
From SOH CAH TOA, TOA states the tangent relationship. $$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}.$$ It compares rise (opposite) to run (adjacent) in a right triangle.
Sample question
Express $y=2\cos\!\left(x+\tfrac{\pi}{2}\right)$ in terms of $\sin x$.
- A$2\sin x$Answer option
- B$-2\cos x$Answer option
- C$2\cos x$Answer option
- D$-2\sin x$Correct answer
Why this answer is right
Use the shift identity $\cos\!\left(x+\tfrac{\pi}{2}\right)=-\sin x$. Thus $y=2\cdot(-\sin x)=-2\sin x$.
The angle addition identity gives $$\cos\!\left(x+\tfrac{\pi}{2}\right)=\cos x\cos\tfrac{\pi}{2}-\sin x\sin\tfrac{\pi}{2}=0-\sin x=-\sin x.$$ Multiplying by $2$ yields $$y=2\cos\!\left(x+\tfrac{\pi}{2}\right)=-2\sin x.$$
Sample question
Solve for all real $x$: $\cos 2x=\sin x$.
- A$x=\tfrac{\pi}{6}+2k\pi$ or $x=\tfrac{5\pi}{6}+2k\pi$Answer option
- B$x=k\pi$Answer option
- C$x=\tfrac{\pi}{6}+2k\pi$, $x=\tfrac{5\pi}{6}+2k\pi$, or $x=\tfrac{3\pi}{2}+2k\pi$Correct answer
- D$x=\tfrac{\pi}{2}+k\pi$Answer option
Why this answer is right
Rewrite $\cos 2x=1-2\sin^2 x$. Solve $1-2\sin^2 x=\sin x$, i.e. $2y^2+y-1=0$ with $y=\sin x$. Then $y=\tfrac{1}{2}$ or $y=-1$, so $x=\tfrac{\pi}{6},\tfrac{5\pi}{6}$ or $x=\tfrac{3\pi}{2}$ modulo $2\pi$.
Use the double-angle identity $\cos 2x=1-2\sin^2 x$ and set $y=\sin x$.
How this page fits into Clevolab
Clevolab is broader than any one exam mode. GCSE and A-level pages are useful entry points, while the wider project is about sharpening understanding through repeated topic practice.