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IB Maths AA: Geometry and trigonometry — Practice Questions & Answers
A complete tour of IB AA Topic 3: 3D coordinate geometry, surface area and volume of solids, right-angled trigonometry, the unit circle with radian measure, arc length and sector area, exact values, the Pythagorean and double-angle identities, the sine and cosine rules with the area formula, solving trigonometric equations, transformations of trig graphs, and the HL vectors toolkit (dot and cross products, lines and planes).
50 practice questions available for this topic — here are 10 with full answers and explanations.
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Practice questions with answers
1
Multiple choice · Easy
What is the distance between the points $A(1,2,3)$ and $B(4,6,3)$?
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Why Distance $=\sqrt{(4-1)^{2}+(6-2)^{2}+(3-3)^{2}}=\sqrt{9+16+0}=\sqrt{25}=5$.
2
Multiple choice · Easy
What are the coordinates of the midpoint of the segment joining $P(2,0,-4)$ and $Q(6,8,2)$?
$(4,4,-1)$
$(4,8,-2)$
$(8,8,-2)$
$(2,4,3)$
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Why Midpoint $=\left(\frac{2+6}{2},\frac{0+8}{2},\frac{-4+2}{2}\right)=(4,4,-1)$.
3
Multiple choice · Easy
A sphere has radius $3$ cm. What is its volume?
$36\pi$ cm$^{3}$
$27\pi$ cm$^{3}$
$12\pi$ cm$^{3}$
$108\pi$ cm$^{3}$
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Why Volume of a sphere $=\frac{4}{3}\pi r^{3}=\frac{4}{3}\pi (3)^{3}=36\pi$ cm$^{3}$.
4
Multiple choice · Easy
In a right-angled triangle the side opposite angle $\theta$ is $3$ and the hypotenuse is $5$. What is $\sin\theta$?
$\frac{3}{5}$
$\frac{4}{5}$
$\frac{3}{4}$
$\frac{5}{3}$
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Why $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{3}{5}$.
5
Multiple choice · Easy
Convert $180^{\circ}$ to radians.
$\pi$
$\frac{\pi}{2}$
$2\pi$
$\frac{\pi}{4}$
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Why Since $180^{\circ}=\pi$ radians by definition, the answer is $\pi$.
6
Multiple choice · Medium
A circular sector has radius $r=6$ and central angle $\theta=\frac{\pi}{3}$ radians. What is the arc length?
$2\pi$
$\pi$
$6\pi$
$\frac{\pi}{2}$
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Why Arc length $=r\theta=6\times\frac{\pi}{3}=2\pi$.
7
Multiple choice · Easy
What is the exact value of $\cos 60^{\circ}$?
$\frac{1}{2}$
$\frac{\sqrt{3}}{2}$
$\frac{\sqrt{2}}{2}$
$1$
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Why From the standard exact values, $\cos 60^{\circ}=\frac{1}{2}$.
8
Multiple choice · Medium
Using $\sin^{2}\theta+\cos^{2}\theta=1$, if $\cos\theta=\frac{4}{5}$ and $\theta$ is acute, what is $\sin\theta$?
$\frac{3}{5}$
$\frac{1}{5}$
$\frac{4}{5}$
$\frac{9}{25}$
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Why $\sin^{2}\theta=1-\left(\frac{4}{5}\right)^{2}=1-\frac{16}{25}=\frac{9}{25}$, so $\sin\theta=\frac{3}{5}$ for acute $\theta$.
9
Multiple choice · Easy
What is the period of the function $y=\sin x$ (with $x$ in radians)?
$2\pi$
$\pi$
$\frac{\pi}{2}$
$4\pi$
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Why The sine function repeats every $2\pi$ radians, so its period is $2\pi$.
10
Multiple choice · Medium
A cone has base radius $r=4$ cm and slant height $l=5$ cm. What is its total surface area?
$36\pi$ cm$^{2}$
$20\pi$ cm$^{2}$
$56\pi$ cm$^{2}$
$24\pi$ cm$^{2}$
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Why Total surface area $=\pi r^{2}+\pi r l=\pi(16)+\pi(4)(5)=16\pi+20\pi=36\pi$ cm$^{2}$.
Key terms in Geometry and trigonometry
Point: A location in space with no size, described by coordinates.
Line segment: The straight part of a line between two endpoints.
Distance formula: The length between two points $(x_1, y_1)$ and $(x_2, y_2)$, equal to $\sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$.
Midpoint: The point halfway between two points, with coordinates the averages of theirs.
Gradient of a line: The measure of steepness equal to the change in $y$ over the change in $x$.
Equation of a line: An algebraic relation describing all points on a straight line.
Parallel lines: Lines in a plane that never meet and have equal gradients.
Perpendicular lines: Lines that meet at a right angle, with gradients whose product is $-1$.
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