Trigonometry tutorial
1. Find the distance 𝑑 in the figure below.
2. The diagram shows the paths in a park.
ˆ E  37 and BDˆ C  56 . 𝐵𝐷 = 420 𝑚
𝐴𝐵𝐶 is a straight line. EAˆ D  ABˆ D  90 , AD
and 𝐴𝐷 = 550 𝑚.
Calculate
(a) AB,
(b) BC,
(c) DE.
Angles of elevation and depression
Angle of elevation
Angle of elevation of B from A
Angle of depression
Angles of elevation and depression
Questions
1. A man 1.8 m tall observes that the angle of elevation of the top of a tree 12 m distant is
32°. What is the height of the tree?
Solution
BE = DC = 12 m (opp. sides of a rectangle)
In ∆𝐴𝐵𝐸
tan 32° =
𝐴𝐵
12
AB = 12 tan 32°
AB = 12 × 0.625
AB = 7.50 m
BC = DE = 1.8 m (opp. sides of a rectangle)
AC = AB + BC
= (7.50 + 1.8) m
= 9.30 m (3 s.f.)
Hence the height of the tree is 9.3 m
2. A man 1.8 m tall is 12 m away from a tree 7.2 m high. What is the angle of elevation of
the top of the tree from his eyes?
3. A man on top of a cliff 80 m high observes that the angle of depression of a buoy at sea
is 12° . How far is the buoy from the cliff?
Solution
PRˆ Q  12 (alternate to 12°)
In ∆𝑃𝑄𝑅
80
tan 12 
PR
80
PR 
tan 12
80
PR 
0.213
𝑃𝑅 = 376𝑚 (3 s.f.)
 the buoy is 376 m from the cliff.
4. A man on top of a cliff 100 m high is in line with two buoys whose angles of depression
are 17° and 19°. Find the distance between the buoys.
Solution
5. K and L are two points in line with a tree such that KL = 20 m. The angles of elevation of
the top of a tree from K and L are 17.5° and 19° respectively. Find the height of the tree.
Solutions to bearing questions
(i)
(ii)
(iii)
Bearing of P from O = 060°
Bearing of Q from O = 60° + 50°
= 110°
Bearing of S from O = 360° − 15°
= 345°
∝= 65° (alternate to 065°)
Bearing of Q from P = 180°+∝
= 180° + 65°
= 245°
(a) Bearing of Q from P = 180° + 40°
= 220°
(b) ∝= 40° (alternate to 40°)
∴bearing of P from Q = 040°
(c) 𝑃𝑄̂ 𝑅 = 𝑃𝑅̂ 𝑄 (equal ∠𝑠 of an isos. ∆)
180°−40°
𝑃𝑅̂ 𝑄 =
2
= 70°
∴ bearing of Q from R = 360° − 70°
= 290°
In ∆𝐴𝐵𝐶
sin 𝐴𝐵̂ 𝐶 =
6
7
sin 𝐴𝐵̂ 𝐶 = 0.857
𝐴𝐵̂ 𝐶 = 59.0° (1 d.p.)
Hence the bearing of C from B =270° − 59°
= 211°
Remaining questions are to be completed by students