Download Triangle Trig - Mr. Lee`s Science

MCR3U – Trig Functions
493720309
Date: __________________________
Triangle Trigonometry
What does SOHCAHTOA mean to you?
For A :
For C
sin A =
sin C =
cos A =
cos C =
tan A =
tan C =
Example:
Find the missing side length x along with the missing angle.
Example:
Find the missing angles.
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MCR3U – Trig Functions
493720309
SOHCAHTOA requires that we have a right angle triangle but what do we do when the
triangle does not have a right angle?
We need to use the Sine Law. From the following triangle:
sin A sin B sin C
a
b
c




or
a
b
c
sinA sinB sinC
Example:
Solve the following triangle. Determine all missing sides and angles.
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MCR3U – Trig Functions
493720309
Can the "c" be solved using Sine Law? ______________
Why or why not? _______________________________________________________
Since we can’t use Sine Law, we have to use Cosine Law.
a 2  b 2  c 2  2bccos A
b 2  a 2  c 2  2accos B
c 2  a 2  b 2  2abcos C
Example:
Find the measure of angle A.
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MCR3U – Trig Functions
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Example:
Given the ΔABC with a = 3, b = 5, A = 30°, determine the measure of angle B.
The Sine Law – The Ambiguous Case
1. Evaluate the following correct to 4 decimal places:
a) sin 65°
sin 115°
b) sin 100°
sin 80°
c) sin 50°
sin 130°
d) sin 170°
sin 10°
e) sin 70°
sin 110°
f) sin 91°
sin 89°
2. What do you notice about the sum of each pair of angles? ____________________
3. What do you notice about the value of each pair? Why? (hint - see diagram below)
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
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MCR3U – Trig Functions
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4. Find the values of theta, , between 0° and 180°, correct to 1 decimal place.
a) sin  = 0.5664
_______________
b) sin  = 0.2598
_______________
c) sin  = 0.8750
_______________
d) sin  = 0.9060
_______________
5. How many possible angles are there? ________________
Examples:
Solve these triangles - you will either get two triangles, one triangle or no triangles
Triangle 1
A = 42°
a = 30
b = 25
Triangle 2
B = 27°
b = 25
a = 30
Triangle 4
D = 30°
d = 50
a = 25
Triangle 3
C = 37.3°
c = 85
b = 90
Triangle 5
E = 38.7°
e = 10
b = 25
Triangle 1
Triangle 2
Triangle 3
Triangle 4
Triangle 5
1 case
2 cases
2 cases
1 case
No solutions
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MCR3U – Trig Functions
493720309
How do you recognize when you need to consider the Ambiguous Case?
To solve an oblique triangle, you need to know the measure of at least one side and any
two other parts of the triangle. There are four cases in which this can happen.
Given Information
What Can Be Found
Law Required
Two angles and any side (AAS or
ASA)
Side
sine law
Two sides and the contained angle
(SAS)
Side
cosine law
Three sides (SSS)
Angle
cosine law
Two sides and an angle opposite one
of them (SSA)
Angle
sine law
Case 4 is called the Ambiguous Case because sometimes it is possible to draw more
than one triangle for the given information.
Hints for recognizing the ambiguous case:
Swing side a as a pendulum fixed at vertex C. Each time line a touches c, you create a
new triangle.
Given A, if a > b then there is only one triangle created.
Given A, if a < b then there are three possible outcomes (Note: h = b sin A)
Homework:
Page 318
Questions 1, 4
Page 325
Questions 1, 3
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