Didactic Lesson

TRIGONOMETRY

Let’s think this through… Is there a unique triangle with the given angle and side

measures? Why? 

Let’s think this through… How might you determine the measures of the missing

angle and sides?  

Discussion

The triangle is unique

AAS is one method for proving that triangles are congruent.

Therefore, if two angles and a side are known, the triangle is unique

Discussion cont’d

Since the sum of the three angles in every triangle is 180°, subtract the two angle measures from 180° to determine the measure of the third angle.

Finding the side lengths is more difficult…

Finding a method for determining the side lengths (or angles) of a unique triangle is the purpose of this lesson.

Lesson Objectives

Use right triangle trigonometry to develop the sine rule

Use the sine rule to solve problems.

Student activity Instructor explanation

An altitude of a triangle extends from a vertex to the opposite side and forms a right angle with the opposite side.

Drawing an altitude of triangle ABC creates two right triangles.

Student activity cont’d

Instructor explanation Since two right triangles are created, right

triangle trigonometry can be used to describe the relationships between the angles and sides of each triangle.

Because triangle ABC shares angles and sides with the two right triangles, the relationships between the angles and sides of the right triangles can be used to describe the relationships between the angles and sides of triangle ABC.

Student activity cont’d

Equations deduced

Student activity (part 2) Instructor facilitation

Students are to follow questions 7 to 13 and derive an equation for the oblique triangle ABC

Students then discusses their results with the instructor

--------(2)

Student Activity cont’d

Connecting equations

Relating equation (1) and (2)

Sine Rule

Applying the sine rule

Students are to use the sine rule to solve for the measure of c and a and angle at A in the triangle discussed at the beginning of the lesson

That is,

Applying the sine rule cont’d The solution to the triangle is shown below

Exercises Students are to answer the questions on the

worksheet. Questions:

Refer to ΔABC, which is not drawn to scale, to answer the following questions

1. Use the sine rule to determine the missing angles and sides if ∠A = 41°, a = 24, and b = 10.

Exercises

2. Use the sine rule to determine the missing angles and sides if ∠A = 32°, a = 6.5 and b = 9.2.

3. Use the law of sines to determine the missing angles and sides if ∠B = 58°, a = 5, and b = 3.4.

Solutions to Exercise [There is only one possible triangle. Although

two values, 15.86° and 164.14°, result for the measure of ∠B, the second answer is impossible. Therefore, ∠b = 15.86, ∠C = 123.14°, and c = 30.63]

There are two possible triangles, because the measure of ∠B could be either 48.59° or 131.41°. If ∠B = 48.59°, then ∠C = 99.41° and c = 12.1. If ∠B = 131.41°, then ∠C = 16.59° and c = 3.5.

There are no solutions, because the law of sines would yield that sin A = (5 × sin 58°) / 3.4 = 1.2471, which is impossible.]

Conclusion

Instructor describes the importance of the sine rule, in that it can be used to solve problems involving non‑right triangles.

Instructor also emphasize that the sine rule cannot solve all problems involving non‑right triangles. E.g. If two angles and a side or two sides and a

non‑included angle of a triangle are known, the law of sines can be used to determine the missing angles and sides of the triangle.

THANK YOU