3.8 Problem Solving Using Trigonometric Ratios

Solving a Right Triangle

To solve a triangle, all unknown angles and all unknown sides need to be found. This can be done by using trigonometric ratios and other properties related to right triangles that were covered throughout this unit.

Practice 1

Solve the right triangles. Round values to one decimal place where necessary.

a)

Right triangle CBA. C is the right angle. Side a is 8 and angle A is 42.

 b)

Right triangle EFD. E is the right angle. Side e is 9 and side d is 4.

 

 

Trigonometry Word Problems

Practice 2

Answer the following questions. Be sure to make note of the units in each question.

a) Pola is planning to put a path of paving stones through her flower garden. The flower garden is a square with sides of 10 feet. What will the length of the path be in metres? Round your answer to two decimal places.

A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.
b) A ladder leans against a brick wall. The foot of the ladder is 3 m from the wall. To the nearest whole number, how long is the ladder if the ladder makes an angle of 50° with the ground?

 

 

 
c) Paula has let 25 m of string out on her kite. She is flying it 11.5 m above her eye level. Find the angle of elevation of the kite. Round to the nearest whole number.

A right triangle with a hypotenuse of 25 m and opposite side is 11.5 m
d) An equilateral triangle has a height of 12 mm. Find the length of each side. Express your answer in centimetres, and round to the nearest hundredth.

 

 
e) Zola is standing 75 feet away from the base of the Living Shangri-La, the tallest building in British Columbia. She looks up to the top of the building at a 83.5° angle. Rounded to the nearest tenth, how tall is the Living Shangri-La?

 

 

 
f) Sunil is standing on the ground, 25 m from the base of a cliff and looks up at his friend, Amelia, on the top of a cliff 0.1 km above the ground. What is the angle of elevation, rounded to the nearest whole number?

 

 

 
g) From the top of a rock wall, the angle of depression to a swimmer is 56°. If the wall is 20 m high, how far from the base of the wall is the swimmer?

 

 

 

Homework

  1. Solve the right triangles. Round values to one decimal place where necessary.

    a)
    Right triangle ACB. C is the right angle. Side a is 6 and angle B is 69.    		 b)
    Right triangle ACB. C is the right angle. Side b is 10 and angle B is 74.
    c)
    Right triangle FED. E is the right angle. Side f is 16 and side d is 9.    		 d)
    Right triangle FED. E is the right angle. Side f is 7 and side e is 10.
    e)
    Right triangle CDB. D is the right angle. Side d is 44 and angle C is 29.    		 f)
    Right triangle TRS. R is the right angle. Side r is 25 and side t is 15.

  2. Amsale wants to attach a 17-foot string of lights to the top of the 15-foot mast of her sailboat as shown in the diagram below. How far from the base of the mast should she attach the end of the light string? Express your answer in inches.

    A picture of a boat is shown. The height of the centre pole is labeled 15 feet. The string of lights is at a diagonal from the top of the pole and is labeled 17 feet.

  3. A ladder leans against the side of a house. If the foot of the ladder is 265 cm from the wall, and the ladder is 7.25 m tall, what angle does the ladder make with the ground? Round to the nearest whole number.

     

     

     

  4. Katherine is in an hot air balloon a vertical distance of 18 m off the ground. If Katherine’s bike is a straight line distance of 40 m from where she is standing in the hot air balloon, what is the angle of depression between her and the bike? Round to the nearest tenth.

     

     

     

  5. From the top of a cliff with a height of 55 m, the angle of depression to a wolf is 30°. To the nearest metre, how far away is the wolf from the top of the cliff.

     

     

     

  6. Monika wants to put a banner across her garage door to congratulate her son on his college graduation. The garage door is 12 feet high and 4.88 metres wide. How long should the banner be to fit the garage door? Round to the nearest whole number.

     

     

     

  7. A tree that is 252 inches tall casts a 40-foot shadow. Rounded to the nearest whole number, at what angle does the sun hit the tree?

     

     

     

  8. Silas, who is 1.73 m tall, is standing 15 m from a building. He sights the top of the building with an angle of elevation of 72°. Find the height of the building. Round to the nearest tenth.

     

     

     

  9. A city engineer plans to build a footbridge across a lake from point X to point Y, as shown in the picture below. To find the length of the footbridge, she draws a right triangle XYZ, with a right angle at X. She measures the distance from X to Z as 800 feet, and from Y to Z, 1 000 feet. How long will the bridge be?

    A lake is shown. Point Y is on one side of the lake, directly across from point X. Point Z is on the same side of the lake as point X.

  10. From the top of a 175 ft lighthouse on the beach, a person sights two whales. The angle of depression to one whale is 12° and to the other whale is 10°. Which whale is closer to the lighthouse and by how much? Round to the nearest tenth.

     

     

     

Answers

1.

a)

\angle A = 21°

\angle B = 69°

\angle C = 90°

a = 6

b = 15.6

c = 16.7

 b)

\angle A = 16°

\angle B = 74°

\angle C = 90°

a = 2.9

b = 10

c = 10.4

 c)

\angle D = 29.3°

\angle E = 90°

\angle F = 60.7°

d = 29.4

e = 18.4

f = 60.6
d)

\angle D = 45.6°

\angle E = 90°

\angle F = 44.4°

d = 7.1

e = 10

f = 7

 e)

\angle B = 61°

\angle C = 29°

\angle D = 90°

b = 38.5

c = 21.3

d = 44

 f)

\angle T = 36.9°

\angle R = 90°

\angle S = 53.1°

t = 15

r = 25

s = 20
2. 96 in 3. 69° 4. 26.7° 5. 110 m 6. 20 ft 7. 28° 8. 47.9 m 9. 600 ft

10. The whale at an angle of depression of 12° is closer to the lighthouse. The whale is 169.2 ft closer.

 

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resources:

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

Wang, M. (2018). Key concepts of intermediate level math. BCcampus. https://collection.bccampus.ca/textbooks/key-concepts-of-intermediate-level-math-bccampus-204/, licensed under CC BY 4.0

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Math 10C Workbook Copyright © 2026 by Leanne Thompson is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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