Unit 15 Section 3 : Calculating Sides

In this section we use the trigonometric functions to calculate the lengths of sides in a right-angled triangle.

Trigonometric Functions

sinΘ  =  cosΘ  =  tanΘ  = 

Example 1

Calculate the length of the side marked x in this triangle.

In this question we use the opposite side and the hypotenuse. These two sides appear in the formula for sinΘ , so we begin with,

sinΘ =

In this case this gives,

sin 40° =
or
x = 8 × sin 40°
= 5.142300877 cm
= 5.1 cm to 1 decimal place

Example 2

Calculate the length of the side AB of this triangle.

In this case, we are concerned with side A B which is the opposite side and side BC which is the adjacent side, so we use the formula,

tanΘ =

For this problem we have,

tan 50° =
so
x = 9 × tan 50°
= 10.72578233 cm
= 10.7 cm to 1 decimal place

Example 3

Calculate the length of the hypotenuse of this triangle.

In this case, we require the formula that links the adjacent side and the hypotenuse, so we use cosΘ .

Starting with

cosΘ =

we can use the values from the triangle to obtain,

cos 20° =
H × cos 20° = 12
H =
= 12.77013327 cm

Therefore the hypotenuse has length 12.8 cm to 1 decimal place.

Exercises

Question 1

Use the formula for the sine to determine the length of the side marked x in each of the following triangles. In each case, give your answer correct to 1 decimal place.

(a)
cm
(b)
cm
(c)
cm
(d)
cm
Question 2

Use the formula for the cosine to determine the length of the adjacent side in each of the following triangles. Give your answers correct to 1 decimal place.

(a)
cm
(b)
cm
(c)
cm
(d)
cm
Question 3

Calculate the length of sides indicated by letters in each of the following triangles. Give each of your answers correct to 3 significant figures.

(a)
x = cm y = cm
(b)
a = m b = m
(c)
x = cm
(d)
p = cm
(e)
z = cm
(f)
x = m
Question 4

Calculate the length of the hypotenuse of each of the following triangles. Give each of your answers correct to 3 significant figures.

(a)
cm
(b)
cm
(c)
cm
(d)
cm
Question 5

Calculate all the lengths marked with letters in the following triangles. Give each of your answers correct to 2 decimal places.

(a)
a = cm b = cm
(b)
c = cm d = cm
(c)
e = cm f = cm
(d)
g = cm h = cm
Question 6

A ladder, which has length 6 m, leans against a vertical wall. The angle between the ladder and the horizontal ground is 65°.

(a)

How far is the foot of the ladder from the wall?

Distance from foot of the wall = m (to the nearest cm)
(b)

What is the height of the top of the ladder above the ground?

Height of top above the ground = m (to the nearest cm)
Question 7

A boat sails 50 km on a bearing of 070°.

(a)

How far east does the boat travel?

Distance east = km (to the nearest km)
Distance east = 50 × sin70° = 46.98463104 km = 47 km (to the nearest km)
(b)

How far north does the boat travel?

Distance north = km (to the nearest km)
Distance east = 50 × cos70° = 17.10100717 km = 17 km (to the nearest km)
Question 8

Calculate the perimeter and area of this triangle.
Give your answers correct to 2 decimal places.

Perimeter =

Area =

Question 9

A ramp has length 6 m and is at an angle of 50° above the horizontal. How high is the top of the ramp? Give your answer to a sensible level of accuracy.

Height of top of ramp = m
Question 10

A rope is stretched from a window in the side of a building to a point on the ground, 6 m from the base of the building. The angle between the rope and the side of the building is 19°.

(a)

How long is the rope?

Length of rope = m (to the nearest cm)
6 ÷ sin19° = 18.42932092 m = 18.43 m (to the nearest cm)
(b)

How high is the window?

Height of window = m (to the nearest cm)
6 × tan19° = 17.42526527 m = 17.43 m (to the nearest cm)