Trig Test Review
[91 marks]
Three airport runways intersect to form a triangle, ABC. The length of AB is 3.1
km, AC is 2.6 km, and BC is 2.4 km.
A company is hired to cut the grass that grows in triangle ABC, but they need to
know the area.
1a. Find the size, in degrees, of angle BAC.
[3 marks]
1b. Find the area, in km2, of triangle ABC.
[3 marks]
The volume of a hemisphere, V, is given by the formula
V = √ 243 ,
4 S3
π
where S is the total surface area.
The total surface area of a given hemisphere is 350 cm2.
2a. Calculate the volume of this hemisphere in cm3.
[3 marks]
Give your answer correct to one decimal place.
2b. Write down your answer to part (a) correct to the nearest integer.
[1 mark]
2c. Write down your answer to part (b) in the form a × 10k , where 1 ≤ a < [2 marks]
10 and k ∈ Z.
A solid glass paperweight consists of a hemisphere of diameter 6 cm on top of a
cuboid with a square base of length 6 cm, as shown in the diagram.
The height of the cuboid, x cm, is equal to the height of the hemisphere.
3a. Write down the value of x.
[1 mark]
3b. Calculate the volume of the paperweight.
[3 marks]
3c. 1 cm3 of glass has a mass of 2.56 grams.
[2 marks]
Calculate the mass, in grams, of the paperweight.
A solid right circular cone has a base radius of 21 cm and a slant height of 35 cm.
A smaller right circular cone has a height of 12 cm and a slant height of 15 cm,
and is removed from the top of the larger cone, as shown in the diagram.
4a. Calculate the radius of the base of the cone which has been removed.
[2 marks]
4b. Calculate the curved surface area of the cone which has been removed. [2 marks]
4c. Calculate the curved surface area of the remaining solid.
[2 marks]
Emily’s kite ABCD is hanging in a tree. The plane ABCDE is vertical.
Emily stands at point E at some distance from the tree, such that EAD is a straight
line and angle BED = 7°. Emily knows BD = 1.2 metres and angle BDA = 53°, as
shown in the diagram
5a. Find the length of EB.
[3 marks]
T is a point at the base of the tree. ET is a horizontal line. The angle of elevation of
A from E is 41°.
5b. Write down the angle of elevation of B from E.
[1 mark]
5c. Find the vertical height of B above the ground.
[2 marks]
Günter is at Berlin Tegel Airport watching the planes take off. He observes a plane that is at an angle
of elevation of
20∘ from where he is standing at point G . The plane is at a height of 350 metres. This
information is shown in the following diagram.
6a. Calculate the horizontal distance, GH, of the plane from Günter. Give
your answer to the nearest metre.
[3 marks]
6b. The plane took off from a point T, which is 250 metres from where
Günter is standing, as shown in the following diagram.
[3 marks]
Using your answer from part (a), calculate the angle ATH, the takeoff angle of the
plane.
The lengths of two of the sides in a triangle are 4 cm and 5 cm. Let θ be the angle
between the two given sides. The triangle has an area of
7a.
Show that sin θ =
5√15
2
2 cm .
[1 mark]
√15
.
4
7b. Find the two possible values for the length of the third side.
[6 marks]
8. Given that sin x = 1 , where 0
3
[6 marks]
2
<x<
π
2 , find the value of cos 4x.
⩽
⩽
9. Solve the equation
Let
sin θ =
sec2 x + 2 tan x = 0, 0 ⩽ x ⩽ 2π.
√5
, where
3
[5 marks]
θ is acute.
10a. Find
cos θ.
[3 marks]
10b. Find
cos 2θ.
[2 marks]
The following diagram shows a triangle ABC and a sector BDC of a circle with
centre B and radius 6 cm. The points A , B and D are on the same line.
^ C is obtuse.
AB = 2√3 cm, BC = 6 cm, area of triangle ABC = 3√3 cm2 , AB
11a. Find
^ C.
AB
[5 marks]
[3 marks]
11b. Find the exact area of the sector BDC.
12. The following diagram shows a sector of a circle where
radians and the length of the arc AB = 2x cm .
Given that the area of the sector is 16
^ =x
AOB
[4 marks]
cm2 , find the length of the arc AB.
At Grande Anse Beach the height of the water in metres is modelled by the
function h(t) = p cos(q × t) + r, where t is the number of hours after 21:00 hours
on 10 December 2017. The following diagram shows the graph of h , for
0 ⩽ t ⩽ 72.
The point A(6.25,
the next high tide.
0.6) represents the first low tide and B(12.5, 1.5) represents
13a. How much time is there between the first low tide and the next high
tide?
[2 marks]
13b. Find the difference in height between low tide and high tide.
[2 marks]
13c. Find the value of p;
[2 marks]
13d. Find the value of q;
[3 marks]
13e. Find the value of r.
[2 marks]
13f. There are two high tides on 12 December 2017. At what time does the
second high tide occur?
[3 marks]
The depth of water in a port is modelled by the function d(t)
0 ⩽ t ⩽ 12, where t is the number of hours after high tide.
= p cos qt + 7.5, for
At high tide, the depth is 9.7 metres.
At low tide, which is 7 hours later, the depth is 5.3 metres.
14a. Find the value of p.
[2 marks]
14b. Find the value of q.
[2 marks]
14c. Use the model to find the depth of the water 10 hours after high tide.
[2 marks]
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