didactic lesson sine rule
TRANSCRIPT
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.
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