Trigonometry - worked solutions

Applied Mathematics 1

Trigonometric Ratios with solutions

O
A
O cos θ = tan θ =
H
H
A
where O = opposite, A = adjacent and H = hypotenuse. sin θ =

1. The angle of elevation of the top of a tree from a point on the ground 10m from the base of the tree is 28◦ . What is the height of the tree?
Solution
We need to find the height of tree which is opposite an angle of 28◦ .Thus
O
tan θ =
A
O = A tan θ height of tree = 10 × tan 28◦
=⇒ height of tree = 5.3m 1d.p.
2. Using a surveying instrument 1.6m high, the angle of elevation of the top of a tower is measured to be 55◦ from a point 6m from the base of the tower. How high is the tower?
Solution
The surveying instrument is already at a height of 1.6m. So we need to find the height of the tower using the instrument, then add this to 1.6m.
O
A
O = A tan θ height of tower using instrument = 6 × tan 55◦
=⇒ height of tower using instrument = 4.2m 1d.p. tan θ =

The height of the tower is 4.2m+1.6m=10.2m high.
3. The angle of elevation of the top of a 20m high mast from a point at ground level is
34◦ . How far is the point from the foot of the mast?
Solution
Let y be the distance of the mast from the point on the ground
O
tan θ =
A
20
◦
tan 34 = y 20 y = tan 34◦ y = 6 × tan 55◦
=⇒ y = 29.7m 1d.p.

4. An isosceles triangle has base 8cm and sloping sides both with length 10cm. What is the base angle of this triangle? Solution
Split the base of the isosceles triangle in half. We now have a right-angled triangle of base 4cm and hypotenuse 10cm. We can now find the angle θ as follows A
H
4 cos θ =
10
4
= 66.4218... = 66◦ to the nearest degree.
10
cos θ =

θ = cos−1

5. A supporting cable of length 30 m is fastened to the top of a 20m high mast. What angle does the cable make with the ground? How far away from the foot of the mast is it anchored to the ground?
Solution
We can use Pythagoras’ Theorem but we shall use trigonometry here. We first find the