1. Module 7

1.36. Page 3

Mathematics 10-3 Module 7 Lesson 7

Module 7: Trigonometry

 

Bringing Ideas Together

 

In Explore you applied the definitions of sine, cosine, and tangent to find missing sides and angles.

 

 

This illustration shows right triangle ABC with angle C as the right angle. BC is a vertical side, is coloured orange, and is labelled side opposite. AC is horizontal, is coloured blue, and is labelled side adjacent. AB is black and is labelled hypotenuse. Angle A is shown in red.

 

 

 

Did you use SOH-CAH-TOA to help you recall these definitions?

 

solve a right triangle: to find all the missing sides and angles in a right triangle

You will now apply these definitions in problem situations where you have to solve a right triangle.

 

In Example 1 you are given the measure of one side and one angle of a right triangle. From this information you will find the missing sides and angles.

 

Example 1

 

In ΔRST, ∠S = 90°, ∠T = 62°, and s = 8 cm. Solve ΔRST. Round to one decimal place when appropriate.

 

Solution

 

Draw a diagram.

 

 

This illustration shows right triangle TRS with angle S as the right angle. Side TR is labelled s = 8 centimetres. Side RS is labelled t, and side TS is labelled r. Angle T measures 62 degrees.

 

 

∠R

  

∠T

62°

∠S

90°

Side s

8 cm

Side r

  

Side t

  

 

You must find ∠R and sides r and t.

 

Remember that all the angles in a triangle add up to 180°.

 

 

 

Next, find side r.

 

Side r is adjacent to the given angle, ∠T. You know the hypotenuse, s.

 

 

This illustration shows right triangle TRS with angle S as the right angle. Side TR is labelled hypotenuse, and equals 8 centimetres. Side RS is labelled t, and side TS is labelled adjacent and r, as well as being drawn in green. Angle T measures 62 degrees.

 

SOH-CAH-TOA

 

 

 

Side r is approximately 3.8 cm.

 

Lastly, find side t. Side t is opposite to the given angle, ∠T. You know the hypotenuse, s.

 

 

This illustration shows right triangle TRS with angle S as the right angle. Side TR is labelled hypotenuse, and equals 8 centimetres. Side RS is labelled opposite and t as well as being drawn in orange, and side TS is labelled r. Angle T measures 62 degrees.

 

SOH-CAH-TOA

 

 

 

Side t is approximately 7.1 cm.

 

In Example 2 you are given the measure of two sides in a right triangle. From this information you will find the missing side and all the unknown angles.

 

Example 2

 

In ΔDEF, ∠F = 90°, d = 5 cm, and f = 8 cm. Solve ΔDEF. Round to one decimal place when appropriate.

 

View the animated “Solving a Triangle: Triangle DEF Solution”

 

As you may have noticed, there are many ways to solve these problems. Depending on the order you solved for the unknowns, you may have chosen different trigonometry ratios to find the unknowns. Ultimately, whatever methods are used, the final solutions for the unknowns will be the same.