Q1
Answer
Convert $230^{\circ}$ to radians.
$4.01$
Q2
Answer
Convert $2.2\pi$ radians to degrees.
$396^{\circ}$
Q3
Answer
Find the exact values of the following without using a calculator:
  1. $\sin\dfrac{7\pi}{3}$
  2. $\cos\dfrac{11\pi}{4}$
  1. $\dfrac{\sqrt{3}}{2}$
  2. $-\dfrac{\sqrt{2}}{2}$
Q4
Answer
Find the exact value of $\tan-\dfrac{2\pi}{3}$ without using a calculator:
$\sqrt{3}$
Q5
Answer
Find a simplified expression for the following when $x$ is sufficiently small: $$\dfrac{\cos x-1}{\sin 2x}$$
$-\dfrac{x}{4}$
Q6
Answer
Find a simplified expression for the following when $x$ is sufficiently small: $$\dfrac{\sin 4x+x^2}{3x-\tan 2x}$$
$4 + x$
Q7
Answer
The points $A$ and $B$ lie on the circumference of a circle with centre $O$ and radius $8.5$ cm. The point $C$ lies on the major arc $AB$. Given that $ACB = 0.4$ radians, find the length of the minor arc $AB$.
$6.8$
Q8
Answer
A sector of a circle has perimeter twice the length of the arc length. Find the size of the angle, in radians, at the centre of the sector.
$2$
Q9
Answer
A class of 24 children stand in a circle. The arc length between each child is $\frac{3\pi}{2}$ m.
  1. Calculate the diameter of the circle in metres.
  2. The children run around the circle at a rate of 2 complete circles per minute. What is the speed they are running at in km per hour?
  1. $36$
  2. $13.6$
Q10
Answer
The area of a sector of a circle of radius $12$ is $100$. Find the perimeter of the sector.
$\dfrac{122}{3}$
Q11
Answer
Find the percentage error in using small angle approximations to find the value of $\dfrac{\cos x}{\sin x}$ when $x = \dfrac{\pi}{24}$.
$0.287\%$
Q12
Answer
A sector of a circle subtends an angle of $2$ radians at the centre. The area of the sector is $30$. Find the exact value of its perimeter.
$4\sqrt{30}$
Q13
Answer
A sector of a circle of radius $28$ cm has perimeter $P$ and area $A$. Given that $A = 4P$, find the value of $P$.
$78.4$
Q14
Answer
Find an approximation for $4\cos\theta + \cos^2 2\theta$ when $\theta = 3^{\circ}$.
$4.98$
Q15
Answer
$POQ$ is a sector of a circle with an angle at the centre of $0.4$ radians. The circle has a radius of $8$, and $R$ is the midpoint of $OQ$. For the region $S$, find the:
  1. perimeter;
  2. area.
  1. $11.8$
  2. $6.57$
Q16
Answer
In the following diagram, $AOB$ is a sector of a circle with an angle at the centre of $1$ radian, and $AC$ is a tangent to the circle at $A$. The circle has a radius of $10$.
  1. Find the area of region $R$.
  2. Given that $BD$ is parallel to $AC$, find the perimeter of region $S$.
  1. $27.9$
  2. $23.0$
Q17
Answer
Find an expression for the area of a sector of a circle with radius $r$ and arc length $l$.
$\dfrac{1}{2}rl$
Q18
Answer
The equation $$\cos(80x) + 2\sin(80x) = 2$$ has a solution close to $0$. Find, as an exact fraction, an approximate value of this solution.
$\dfrac{2-\sqrt{2}}{80}$