Print this chapterPrint this chapter

Module 7

1. Module 7

1.20. Page 3

Mathematics 10-3 Module 7 Lesson 4

Module 7: Trigonometry

 

Bringing Ideas Together

 

In Explore you looked at indirect measurement and the sine ratio. You used the following relationship to find the height of the selected hill.

 

 

This illustration shows a right triangle ABC with angle A marked in red and angle C marked with a square. Side AC is labelled side adjacent and is green, side BC is labelled side opposite and is orange, and side AB is labelled hypotenuse and is black.

 

 

 

This relationship can be used as you did in Explore to find the side opposite. This relationship can also be used to determine the hypotenuse or the reference acute angle in a variety of problem situations. Study these examples.

 

Example 1

 

A 25-ft-long wheelchair ramp is inclined at an angle of 4.6° to the horizontal. From the foot of the ramp to the top, what is the vertical rise? Round to the nearest tenth.

 

Solution

 

Draw a diagram.

 

 

This illustration shows a right triangle of height x. The horizontal base has angle A measuring 4.6 degrees at the end opposite the right angle. The hypotenuse measures 25 feet.

 

Let x be the vertical rise.

 

Substitute into the formula.

 

 

 

The vertical rise of the ramp is approximately 2.0 ft.

 

Example 2

 

In this photo, a cyclist rides downhill on a mountain bike.

© BORTEL Pavel/shutterstock

A ladder leans against a vertical wall. The foot of the ladder is positioned for safety at an angle of 75° with the ground. The ladder reaches 14.5 ft up the wall. To the nearest foot, what is the length of the ladder?

 

View the animated “Ladder Solution.”

 

Example 3

 

A mountain trail used by cyclists to train on drops 235 m from the trail head to the bottom over a trail distance of 1.1 km. To the nearest degree, what is the average slope of this trail?

 

Solution

 

Do you remember how many metres are in a kilometre? Review “SI Length Conversion” if you need a refresher.

 

Draw a diagram.

 

 

This illustration shows a right triangle with a horizontal base and a vertical height of 235 metres. The angle opposite the right angle is labelled A, and the hypotenuse is 1.1 kilometres, which is 1100 metres long.

 

Let ∠A be the required angle.

 

All distances must be in the same units.

 

 

 

Substitute into the formula.

 

 

 

The average slope of the trail is approximately 12°.

 

Practise these skills!