Download MBF 3C 2007-2008 Lesson 1 Revisit the Primary Trigonometric

MBF 3C
2007-2008
Lesson 1
Revisit the Primary Trigonometric Ratios
The primary trigonometric (“trig”) ratios can only be used with right triangles. They can be
used to find both the sides and the angles of these triangles, given enough information.
Remember that we use capital letters to name angles. The side opposite each of these angles
is named using the corresponding lower-case letter.
A
Hypotenuse
Adjacent
Opposite
C
opposite

hypotenuse
adjacent
cos A 

hypotenuse
opposite
tan A 

adjacent
sin A 
Use
Opposite
Adjacent
B
opposite

hypotenuse
adjacent
cos B 

hypotenuse
opposite
tan B 

adjacent
sin B 
SOH CAH TOA to help you remember the primary trig ratios.
Solving a triangle means that you find all side measures and all angles.
When no rounding instructions are given, always round to match the information given in the
question.
Look at the above diagram and the ratios. Is it possible to solve a triangle when you are given
only one side measure?
Unit 1 – Trigonometry
Page |1
MBF 3C
2007-2008
Example 1
Writing Trig Ratios
Write the trig ratios for sin A, cos A, and tan A for the given triangle. Express each ratio in
lowest terms.
A
26
10
C
B
24
Example 2
Using a Calculator to Determine Trig Ratios
Evaluate, rounding to 4 decimal places.
Sin 52º =
cos 43º =
Example 3
tan 21º =
tan 90º =
Finding an Angle Given its Trig Ratio
Find the measure of angle D (D) if sin D = 0.5971. Round your answer to the nearest tenth
of a degree (1 decimal place).
Example 4
Find the Length of a Side
1.
A
Label the sides of the triangle with respect
to the given angle.
2. Choose the appropriate ratio to determine
the missing side and write it out using the
unknown quantity.
c
10
3. Solve for the unknown.
23 
C
B
a
Unit 1 – Trigonometry
Page |2
MBF 3C
2007-2008
Example 5
Find an Angle Given the Length of Two Sides
Find the measure of A in the following triangle to the nearest hundredth of a degree.
C
13
A
19
B
Example 6
Solve a Right Triangle
Solve the right triangle. Whenever necessary, round to the nearest tenth.
C
B
30 
27 cm
A
Homework:
Pages 13 – 15 # 2 - 12
Unit 1 – Trigonometry
Page |3
MBF 3C
2007-2008
Lesson 2
Solve Problems Using Trigonometric Ratios
The angle of elevation is also sometimes referred to as the angle of inclination. It is the
angle you look up through from the horizontal line of sight.
Angle of
Elevation
Horizontal Line of Sight
The angle of depression is also sometimes referred to as the angle of declination. It is the
angle you look down through from the horizontal line of sight.
Horizontal Line of Sight
Angle of
Depression
Use this strategy when solving problems using trig ratios:
1. Draw a fully labeled diagram.
2. Identify and mark the needed measure using an appropriate variable.
3. Write a “Let . . . . . “ statement to define this variable.
4. Write an equation using a trig ratio substituting in all known and unknown values from
your diagram.
5. Solve for the unknown by manipulating your equation algebraically.
6. Write a concluding statement that answers the question you were asked in the problem.
Unit 1 – Trigonometry
Page |4
MBF 3C
2007-2008
Example 1
Calculating Distances
P. 23 #14
The Instrumental Landing System (ILS) common to most major airports uses radio beams to
bring an aircraft down a 3º glide slope. A pilot noted that his height above the ground
(altitude) was 200 m. How far would the pilot have to travel before landing on the runway?
Example 2
Calculate the Measure of an Angle
P. 23 #13
The shuttle Enterprise lifts off from Cape Canaveral. Calculate the angle of elevation of the
shuttle, from an observer located 8 km away, when the shuttle reaches a height of 3500 m.
Homework:
Pages 21 – 23 # 1 - 12
Unit 1 – Trigonometry
Page |5
MBF 3C
2007-2008
Lesson 3
The Sine Law
An acute triangle is a triangle where all angles measure less than 90º. An acute triangle
can be solved if you know:
 Two angles and one side
 An angle measure and two side
measures as long as one of the
sides is opposite the known angle
The measure of the side of a triangle can be calculated using the proportion known as the Sine
Law:
a
sin A

b
sin B

c
sinC
The measure of an angle is calculated using this form of the Sine Law:
sin A
a

sin B
b

sinC
c
To solve for a missing quantity you only need to use two of the ratios from the Sine Law.
Look carefully at the above information. Is it possible to solve a triangle using the Sine Law
when only given all 3 side measures?
Unit 1 – Trigonometry
Page |6
MBF 3C
2007-2008
Example 1
Find the Measure of a Side
P. 31 #1c
Find the measure of the indicated side.
X
60 cm
Y
55 
47 
x
Example 2
Z
Find the Measure of an Angle
Find the measure of the unknown angle to the nearest tenth of a degree.
X
14.2 cm
Y
10.8 cm
45 
Z
Remember that when you are asked to solve a triangle, you are finished once you
can state the measures of all 3 sides and all 3 angles.
Unit 1 – Trigonometry
Page |7
MBF 3C
2007-2008
Example 3
Solve a Triangle
Solve the following triangles. Round all answers to the nearest unit of measurement.
(a)
X
Y
35 
21 
18 cm
Unit 1 – Trigonometry
Z
Page |8
MBF 3C
2007-2008
(b) ABC, where B = 73º, b = 13 cm and c = 12 cm.
Unit 1 – Trigonometry
Page |9
MBF 3C
2007-2008
Example 4
Solve Problems using The Sine Law
P. 33 #11
The Leaning Tower of Pisa leans 5.5º from its vertical. Suppose that the sun is directly
overhead. A surveyor notices that the distance from the base of the tower to the tip of its
shadow is 5.35 m. What is the height of the tower on the lower side to the nearest tenth of a
metre?
Homework:
Pages 31 – 33 #1ab - 10
Unit 1 – Trigonometry
P a g e | 10
MBF 3C
2007-2008
Lesson 4
The Cosine Law
Consider a strategy for solving the triangle and write it out step by step beside the diagram.
X
22 cm
Y
21 
Z
18 cm
Conclusions?
For any triangle, to find the measure of any given side, given the measures of the other two
sides and their contained angle, the Cosine Law can be written as follows:
A
a2 = b2 + c2 – 2bc cosA
b
b2 = a2 + c2 – 2ac cosB
c
c2 = a2 + b2 – 2ab cosC
C
a
B
When you are given the measure of all 3 sides, you can substitute into the above equations or
use these rearrangements:
cos A 
b 2 c 2 a 2
2bc
Unit 1 – Trigonometry
cos B 
a 2 c 2 b 2
2ac
cos C 
a 2  b 2 c 2
2ab
P a g e | 11
MBF 3C
2007-2008
Example 1
Find the Measure of a Side Given Two Sides
P. 39 #1c
and the Contained Angle
Calculate the value of the unknown.
A
6.5 m
c
28 
B
6.0 m
Example 2
Find the Measure of an Angle Given
P. 39 #2c
Three Sides
Find the measure of angle A.
A
11 km
7 km
9 km
Unit 1 – Trigonometry
B
P a g e | 12
MBF 3C
2007-2008
Example 3
Solve a Problem using The Cosine Law
P. 41 #13
Two ships set sail from port. Alpha is sailing at 12 knots and Beta is sailing at 10 knots. After 3
h, the ships are 24 nautical miles apart. Calculate the angle between the ships at the time they
sailed from the port, to the nearest degree. (1 knot = 1 nautical mile per hour)
Homework:
Pages 39 – 41 #1ab, 2ab, 3 - 10
Unit 1 – Trigonometry
P a g e | 13
MBF 3C
2007-2008
Lesson 5
Making Decisions Using Trigonometry
Whenever you are working on problem-solving, it is very important to read through the
question carefully and thoroughly understand the situation and identify the given information.
It is a good idea to draw a diagram as you are reading through the problem the second time,
marking in known information as you come across it. Once you have the picture, it is much
easier to decide which trig strategy to employ.
If the problem is modeled by a right triangle, use the primary trig ratios.
If the problem is modeled by an acute triangle, you have to decide between the Sine Law and
Cosine Law based on the given information.
Problem Modelled by
Triangle
Right Triangle
sin
cos
Acute Triangle
tan
Cosine Law
Sine Law
2 angles and a given
side
2 sides and the
contained angle
2 sides and an angle
opposite one of them
3 sides
Consider the above strategies.
Is it possible to use any of these strategies to solve a right triangle given the measure of just
one side?
Is it possible to find an angle measure in a triangle given the measures of one angle and one
side?
Unit 1 – Trigonometry
P a g e | 14
MBF 3C
Example 1
2007-2008
Use the Sine Law
From an airplane, a surveyor observes two points, A and B, with A being closer to the plane and
9750 m away in a direct line. If the angle of depression to point A is 45º and the angle of
depression to point B is 32º, how far is it across the lake? *Remember the properties of parallel lines.
Example 2
Use the Cosine Law
A surveyor needs to estimate the length of a swampy area. She starts at one end of the
swamp and walks in a straight line, 450 paces, and turns 60º towards the other end of the
swamp. She then walks in another straight line, 380 paces, before arriving at the other end of
the swamp. One paces is about 75 cm. Estimate the length of the swamp in metres.
Unit 1 – Trigonometry
P a g e | 15
MBF 3C
2007-2008
Example 3
Use Primary Trigonometric Ratios
Josh is building a garden shed that is 4 m wide. The two sides of the roof must meet at an 80º
angle and be equal in length. How long must each rafter be if he allows for a 0.5 m overhang?
Homework:
Pp. 48 – 51 # 2odds, 3, 4, 6 - 11
Unit 1 – Trigonometry
P a g e | 16