SOLVING FOR MISSING SIDES IN RIGHT TRIANGLES WITH TRIGONOMETRY

Set up a trigonometric equation connecting the angle with the sides given :

Problem 1 :

Solution :

a = hypotenuse

x = opposite side

sin θ = opposite side/hypotenuse

sin 68° = x/a

0.927 = x/a

x = 0.927a

Problem 2 :

Solution :

b = hypotenuse

x = adjacent side

cos θ = adjacent side/hypotenuse

cos 37° = x/b

0.798 = x/b

x = 0.798b

Problem 3 :

Solution :

c = adjacent side

x = opposite side

tan θ = opposite/adjacent

tan 58° = x/c

1.6 = x/c

x = 1.6c

Problem 4 :

Solution :

d = adjacent side

x = hypotenuse

cos θ = adjacent/hypotenuse

cos 42° = d/x

0.74 = d/x

x = d/0.74

Problem 5 :

Solution :

e = opposite side

x = adjacent side

tan θ = opposite side/adjacent

tan 51° = e/x

1.23 = e/x

x = e/1.23

Problem 6 :

Solution :

f = opposite side

x = adjacent side

tan θ = opposite side/adjacent

tan 71° = f/x

2.90 = f/x

x = f/2.90

Find, to 2 decimal places, the unknown length in:

Problem 7 :

Solution :

Opposite side = x cm

Hypotenuse = 9 cm

sin θ = opposite/hypotenuse

sin 68° = x/9

0.92 = x/9

x = 0.92 × 9

x = 8.28 cm

Problem 8 :

Solution :

Here, adjacent side = x cm

Hypotenuse = 10 cm

cos θ = adjacent/hypotenuse

cos 37° = x/10

0.79 = x/10

x = 0.79 × 10

x = 7.99 cm

Problem 9 :

Solution :

Here, Opposite side = x cm

Adjacent side = 3.82 cm

tan θ = opposite/adjacent

tan 58° = x/3.82

1.60 = x/3.82

x = 1.60 × 3.82

x = 6.11 cm

Problem 10 :

You are hiking near a canyon. While standing at A, you measure an angle of 90º between B and C, as shown. You then walk to B and measure an angle of 76° between A and C. The distance between A and B is about 2 miles. How wide is the canyon between A and C?

Solution :

Opposite side = AC = x

Adjacent side = AB = 2 miles

tan θ = opposite/adjacent

tan 76° = AC/AB

4.010 = x/2

x = 2(4.010)

x = 8.02 miles

Problem 11 :

A parasailer is attached to a boat with a rope 72 feet long. The angle of elevation from the boat to the parasailer is 28°. Estimate the parasailer’s height above the boat.

Solution :

Opposite side = h

Hypotenuse = 72 ft

sin θ = opposite/hypotenuse

sin 28° = h/72

0.4694 = x/72

x = 0.4694(72)

= 33.79

approximately 33.8 ft

Problem 12 :

To measure the width of a river, you plant a stake on one side of the river, directly across from a boulder. You then walk 100 meters to the right of the stake and measure a 79° angle between the stake and the boulder. What is the width w of the river?

Solution :

tan θ = opposite side / adjacent side

tan 79 = w / 100

w = tan 79(100)

= 5.144(100)

= 514.4 m

So, the width of the river is 515 m.

Problem 13 :

Katoomba Scenic Railway in Australia is the steepest railway in the world. The railway makes an angle of about 52° with the ground. The railway extends horizontally about 458 feet. What is the height of the railway?

Solution :

Let h be the height of the railway.

tan θ = opposite side / adjacent side

tan 52 = h / 458

h = 458(tan 52)

= 458 (1.2799)

= 586.213

So, height of the railway is approximately 586 ft.

Problem 14 :

You are standing on the Grand View Terrace viewing platform at Mount Rushmore, 1000 feet from the base of the monument

a. You look up at the top of Mount Rushmore at an angle of 24°. How high is the top of the monument from where you are standing? Assume your eye level is 5.5 feet above the platform.

b. The elevation of the Grand View Terrace is 5280 feet. Use your answer in part (a) to find the elevation of the top of Mount Rushmore.

Solution :

a) 

Opposite side = b and adjacent side = 1000 ft

tan 24 = b / 1000

b = tan 24 (1000)

= 0.4452(1000)

b = 445.2

Height of the eye level = 5.5 ft

Height of the monument from where you are standing = 445.2 + 5.5

= 450.7 ft

Approximately 451 ft.

b) The elevation of the Grand View Terrace = 5280 feet.

= 451 + 5280

= 5731 ft