SL 3.4—Circle, radians, arcs, sectors Questionbank

Draw a fully-labelled 3-D diagram showing $A,B,P_0,P$ and all known angles.

Let the horizontal distances $AP_0 = x$ m and $BP_0 = y$ m.
Show that
$x\tan28^\circ = y\tan35^\circ.$

Using the cosine rule in triangle $ABP_0$, show that
$80^2 = x^2 + y^2 - 2xy\cos60^\circ.$

Use your results from part 2 and 3 to find the height $h$ of the drone above the ground.
Give your answer to the nearest metre.

The drone then moves horizontally so that its projection $P_0$ traces out a circular path of radius 80 m centred at a fixed point on the ground.
Calculate, in metres and in radians, the length of the arc of this path when the bearing of $P_0$ from $A$ changes from $050^\circ$ to $130^\circ$.

The tension in each cable is proportional to its length.
If the tension in $AP$ is $T_A = k|AP|$ and in $BP$ is $T_B = k|BP|$, find the ratio $\dfrac{T_A}{T_B}$ at this instant.

Sketch a fully-labelled 3-D diagram showing $A,B,P_0,P$ and all given angles.

Let $AP_0=x$ m and $BP_0=y$ m.
Show that
$x\tan32^\circ = y\tan45^\circ.$

Using the cosine rule in triangle $ABP_0$, write down an equation relating $x$ and $y$.

Use your results to find the height $h$ of the helicopter above the ground, correct to the nearest metre.

The point $P_0$ later moves horizontally around a circular path of radius 90 m centred at a base camp.
Find the length of the arc, in metres and radians, when its bearing from $A$ changes from $010^\circ$ to $100^\circ$.

If tensions are $T_A = k|AP|$ and $T_B = k|BP|$, find $\dfrac{T_A}{T_B}$.

Draw a fully-labelled 3-D diagram showing $A,B,P_0,P$ and all the given angles.

Let $AP_0=x$ m and $BP_0=y$ m.
Show that
$x\tan28^\circ = y\tan37^\circ.$

Using the cosine rule in triangle $ABP_0$, write an equation relating $x$ and $y$.

Find the height $h$ of the balloon, to the nearest metre.

The projection $P_0$ moves horizontally on a circle of radius 75 m.
Find, in metres and radians, the length of the arc when the bearing of $P_0$ from $A$ changes from $045^\circ$ to $115^\circ$.

If $T_A = k|AP|$ and $T_B = k|BP|$, find $\dfrac{T_A}{T_B}$.

a. $10^{\circ}$ b. $18^{\circ}$ c. $270^{\circ}$

2.Express the following angles in degrees.

a. $\frac{\pi}{8}$ b. $\frac{\pi}{3}$ c. $3 \pi$

$-720^{\circ}<\theta<-360^{\circ}$ or $-4 \pi<\theta<-2 \pi$

$-360^{\circ}<\theta<0^{\circ}$ or $-2 \pi<\theta<0$

$360^{\circ}<\theta<720^{\circ}$ or $2 \pi<\theta<4 \pi$

$720^{\circ}<\theta<1080^{\circ}$ or $4 \pi<\theta<4 \pi$

Find the length of the minor arc AB and the major arc AB .

Find the area of the minor sector and the major sector.

find the perimeter of the minor sector and the major sector.

Draw a fully-labelled 3-D diagram showing $C_1,C_2,D_0,D$ and all known angles.

Let the horizontal distances $C_1 D_0 = x$ m and $C_2 D_0 = y$ m. Show that $x\tan45^\circ = y\tan60^\circ .$

Using the cosine rule in triangle $C_1C_2D_0$, express $y$ in terms of $x$ and known constants.

Combine your results from parts (b) and (c) to find the height $h$ of the dish. Give your answer to the nearest tenth of a metre.

The dish's projection $D_0$ describes a circular path of radius $x$ m. The path subtends an angle of $2.5$ radians. Calculate the arc length in metres.

Find the ratio of the horizontal distance from $C_2$ to the 3-D distance $|C_2 D|$ at this instant.

Show that the angle $\theta$ is $\frac{20-2 r}{r}$

The area of the sector is $25 \mathrm{~cm}^{2}$. Find the value of $r$.

Calculate the radius of the circular flower bed.

Determine the perimeter of the sector that will be used for seasonal flowers.

Convert $28°$ to radians.

Convert $36°$ to radians.

Convert $67°$ to radians.