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NSWMaths AdvancedQuick questions
Year 12: Trigonometric Functions
Quick questions on Radians, arc length, sector and segment area for HSC Maths Advanced
15short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.
What is definition of radian?Show answer
One radian is the angle subtended at the centre of a circle by an arc of length equal to the radius. Equivalently, the radian measure of an angle is the ratio of arc length to radius:
What is conversion?Show answer
$$180^\circ = \pi \text{ radians}, \qquad 1^\circ = \frac{\pi}{180} \text{ rad}, \qquad 1 \text{ rad} = \frac{180^\circ}{\pi} \approx 57.30^\circ.$$
What is arc length?Show answer
For a sector of radius $r$ with central angle $\theta$ in radians, the arc length is
What is sector area?Show answer
The area of a sector of radius $r$ with central angle $\theta$ in radians is
What is triangle and segment?Show answer
The triangle formed by the two radii and the chord has area
What is chord length?Show answer
By the cosine rule (or by splitting the isosceles triangle), the chord opposite the central angle $\theta$ has length
What is converting?Show answer
$120^\circ = 120 \cdot \frac{\pi}{180} = \frac{2 \pi}{3}$ radians.
What is sector area with angle in degrees?Show answer
A sector has radius $8$ cm and central angle $45^\circ$. Convert: $45^\circ = \frac{\pi}{4}$ rad.
What is segment area?Show answer
Circle of radius $5$ cm, central angle $\frac{2 \pi}{3}$.
What is chord?Show answer
For $r = 7$, $\theta = \frac{\pi}{3}$:
What is using degrees in radian formulas?Show answer
$\ell = r \theta$ and $A = \frac{1}{2} r^2 \theta$ require $\theta$ in radians. Substituting $90$ instead of $\frac{\pi}{2}$ gives a wildly wrong answer.
What is forgetting to subtract the triangle?Show answer
A segment is not the sector; subtract the triangle.
What is wrong formula for the triangle?Show answer
The triangle area is $\frac{1}{2} r^2 \sin \theta$ (with $\sin \theta$, not $\theta$). Confusing this with the sector formula gives a wrong segment.
What is calculator in the wrong mode?Show answer
Always check that your calculator is in radian mode when working from $\frac{\pi}{6}$ etc.
What is confusing the minor and major segment?Show answer
"Minor" is the smaller piece, between the chord and the shorter arc. Make sure $\theta$ refers to the central angle of that smaller piece (less than $\pi$).