Trigonometry

Practise Trigonometry for FBISE Grade 9–10 Mathematics (Grade 9–10): 6 multiple-choice questions with instant checking and worked explanations. Free, no sign-up.

Tap an answer to check it · 6 questions

1. Question 1

   The angle of elevation to a $40\mathrm{m}$ tall tower is $30^{\circ }$. What is the horizontal distance from the observer to the base of the tower (in metres)?
   • A30
   • B$40\sqrt{3}$
   • C$\frac{80}{\sqrt{3}}$
   • D$20\sqrt{3}$
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   Answer: B) $40\sqrt{3}$

   Using the definition of tangent for a right‑triangle, $\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$. With $\theta =30^{\circ }$ and opposite side $40\text{m}$, we get $\frac{1}{\sqrt{3}}=\frac{40}{d}$, giving $d=40\sqrt{3}\text{m}$. The other options arise from common mistakes: treating the angle as $45^{\circ }$, halving the height before applying tangent, or using sine instead of tangent.

2. Question 2

   If $\sin \theta =\frac{3}{5}$ and $\theta$ is acute, what is $\cos \theta$?
   • A$\frac{3}{5}$
   • B$\frac{4}{5}$
   • C$\frac{5}{4}$
   • D1
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   Answer: B) $\frac{4}{5}$

   The Pythagorean identity links sine and cosine for any angle. Substituting the given sine value and solving for cosine yields cosθ = √(1‑(3/5)²) = 4/5. The wrong options arise from typical confusions: thinking cosine equals the given sine, assuming cosine of any acute angle is 1, or inverting the sine value.

3. Question 3

   In a $30^{\circ }$-$60^{\circ }$-$90^{\circ }$ triangle, the ratio of the side opposite $30^{\circ }$ to the hypotenuse is:
   • A$\sqrt{3}$
   • B$\frac{\sqrt{3}}{2}$
   • C1
   • D$\frac{1}{2}$
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   Answer: D) $\frac{1}{2}$

   In a $30^{\circ }$‑$60^{\circ }$‑$90^{\circ }$ right triangle the side lengths follow the well‑known ratio $1:\sqrt{3}:2$. The side opposite the $30^{\circ }$ angle is the shortest side and is exactly half the length of the hypotenuse, giving the ratio $\frac{1}{2}$. The other options are incorrect: $1$ would imply the side equals the hypotenuse, which is impossible; $\frac{\sqrt{3}}{2}$ is the ratio for the side opposite $60^{\circ }$; $\sqrt{3}$ omits the denominator and is therefore an arithmetic slip.

4. Question 4

   In a right triangle, one acute angle is three times another, and the side opposite the larger acute angle measures $12\text{cm}$. What is the length of the side opposite the smaller acute angle (in centimetres, nearest whole number)?
   • A5 cm
   • B7 cm
   • C6 cm
   • D4 cm
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   Answer: A) 5 cm

   In a right triangle, the two acute angles sum to $90^{\circ }$. Letting the smaller angle be $\theta$, we have $\theta +3\theta =90^{\circ }⟹4\theta =90^{\circ }⟹\theta =22.5^{\circ }$. The larger acute angle is $3\theta =67.5^{\circ }$. The side opposite the larger angle is $12\text{cm}$. Using the tangent ratio, $\tan 22.5^{\circ }=\frac{x}{12}$, where $x$ is the side opposite the smaller angle. Solving for $x$ gives $x=12\tan 22.5^{\circ }\approx 12\times 0.4142=4.97\text{cm}$, which rounds to $5\text{cm}$.

5. Question 5

   In a right triangle, if one acute angle is twice the other, what is the measure of the smaller acute angle?
   • A$30^{\circ }$
   • B$60^{\circ }$
   • C$45^{\circ }$
   • D$15^{\circ }$
   Show answer and explanation

   Answer: A) $30^{\circ }$

   If the smaller acute angle is $\theta$, the larger is $2\theta$. Their sum must be $90^{\circ }$, giving $3\theta =90^{\circ }$ and $\theta =30^{\circ }$. The other options stem from common errors: assuming equal acute angles, picking the larger angle, or reversing the given ratio.

6. Question 6

   Which statement about the graph of $y=\sin x$ is true?
   • AIts maximum value is 1
   • BIt is symmetric about the y-axis
   • CIt crosses the x-axis at $90^{\circ }$
   • DIt has a period of $180^{\circ }$
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   Answer: A) Its maximum value is 1

   The sine curve reaches a maximum value of $1$, which occurs at $90^{\circ }$ (and every $360^{\circ }$ thereafter). The other statements are false: sine is odd, its period is $360^{\circ }$, and its zeros are at $0^{\circ },180^{\circ },\dots$.

More FBISE Grade 9–10 Mathematics topics

• Coordinate Geometry (3)
• Linear Equations & Inequalities (6)
• Probability & Statistics (8)
• Quadratic Equations (5)
• Real & Complex Numbers (7)
• Sets, Functions & Groups (8)