Download Trigonometry Student Notes

CHAPTER 2: TRIGONOMETRY
Approx. 8 classes
1. Lesson 1: The Tangent Ratio – pg. 70-83 (1 class)
Assignment: pg. 75-77 #3-11, 13, 14, 16-18, 21
pg. 82-83 #3-11, 14, 16
2. Lesson 2: Math Lab: Measuring an Inaccessible Height – pg. 84-86
(1 class)
Assignment: pg. 86 #1-3
3. Lesson 3: The sine and Cosine Ratios – pg. 89-102 (1 class)
Assignment: pg. 95-96 #1-13, 15,
Assignment: pg. 101-102 #1-10, 12
4. Lesson 4: Applying the Trigonometric Ratios – pg. 105-112 (1 class)
Assignment: pg. 111-112 #3-9, 11-13, 16
pg. 88 #1-5, pg. 104 #1-5
5. Lesson 5: Solving Problems Involving More than One Right
Triangle – pg. 113-121 (1 class)
Assignment: pg. 118-121 #1-9, 11, 13-14, 18, 20
6. Chapter Quiz – (1 class)
7. Chapter Review – pg. 124-126 (1 class)
Assignment: pg. 124-126 #1-9, 11-15, 17-20, 22, 23
8. Chapter Exam – (1 class)
1
LESSON 1: The Tangent Ratio
Learning Outcome: Learn to develop the tangent ratio and relate it to the
angle of inclination of a line segment.
Learning Outcome: Learn to apply the tangent ratio to calculate lengths.
Key Math Learnings: In a right triangle, the ratio of the side opposite an
acute angle to the side adjacent to it depends only on the measure of that
angle.
Key Math Learnings: Because the tangent ratio is constant for a given
angle in a right triangle, it can be applied to calculate the length of a side of
any right triangle, even when we cannot measure the length directly.
Making Connections:
Ask all students to stand up and face a wall. Ask the students to stare
directly at the wall and imagine a straight line from their eyes to the wall,
this imaginary line is called the horizontal. Now ask the students to look up
to where the wall meets the ceiling. The angle made from the horizontal to
the corner of the ceiling is called the angle of inclination.
The angle of inclination of a line or line segment is the acute angle it
makes with the horizontal.
Angle of elevation
We name the sides of a right triangle in relation to one of its acute angles.
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Recall Similar Triangles in Math 9:
Ex. ABC is similar to RPQ . Find the lengths of RP and AC.
A
R
21.0 m
Q
C
8.0
B
12.0
16.8
P
Since similar, the ratios of corresponding sides are:
RP PQ RQ


AB BC AC
Sub in the values given:
RP 16.8 21.0


8.0 12.0 AC
Find RP and AC using the ratios above:
In this chapter, we will again be dealing with ratios.
The ratio:
Length of side opposite ∠A: Length of side adjacent to ∠A depends
only on the measure of the angle, not on how large or small the triangle is.
C
Side opposite ∠A
A
B
Side Adjacent to ∠A
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This ratio is called the tangent ratio of ∠A.
The tangent ratio for ∠A is written as tanA.
We usually write the tangent ratio as a fraction.
For solving triangles, there are set ratios set up for Math. These are used
to solve triangles. These ratios apply to all right angled triangles. The first
ratio we will be discussing is Tangent or Tan.
tan 𝐴 =
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑠𝑖𝑑𝑒
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑠𝑖𝑑𝑒
Ex. Given the following triangle, label the adjacent and opposite sides:
The sides of adjacent and opposite will change depending on where the
angle is given. The hypotenuse side is always the longest side and is not
affected by the location of the given angle. The hypotenuse side is always
opposite the right angle.
Although the tangent of an angle is defined as a ratio, you can think of it as
a number that compares the two shorter sides (or legs) of a right triangle.
For example, if tan = 0.75, this means that the side opposite the angle is
0.75 times as long as the side adjacent to the angle.
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Ex. Given the triangle:
C
5 cm
A
B
12 cm
Find Tan A and Tan C ratios:
You can use a scientific calculator to determine the measure of an acute
angle when you know the value of its tangent. The 𝑡𝑎𝑛−1 or InvTan
calculator operation does this.
Refer to the last example. If we want to figure out the angle created by
Tan A and Tan C, we would need to use the 𝑡𝑎𝑛−1 .
𝑇𝑎𝑛 𝐴 ∶ 𝑡𝑎𝑛−1 (
𝑇𝑎𝑛 𝐵: 𝑡𝑎𝑛−1 (
5
5
)=
12
12
)=
5
Ex. A small boat is 95m from the base of a lighthouse that has a height of
36m above sea level. Calculate the angle from the boat to the top of the
lighthouse. Express your answer to the nearest degree.
36 m
95 m
Ex. A 10-ft ladder leans against the side of a building with its base 4ft. from
the wall. What angle does the ladder make with the ground?
Need to find out length of wall, use Pythagorean theorem:
10ft
4 ft
6
The tangent ratio is a powerful tool we can use to calculate the length of a
leg of a right triangle. We are then measuring the length of a side of a
triangle indirectly. In a right triangle, we can use the tangent ratio,
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
,
to write an equation. When we know the measure of an acute angle and
the length of a leg, we solve the equation to determine the length of the
other leg.
Note:
We use direct measurement when we use a measuring instrument to
determine a length or an angle in a polygon. We use indirect measurement
when we use mathematical reasoning to calculate a length or an angle.
To determine tangents of angles, you need a scientific calculator. Your
calculator must be in degree mode.
Ex. Enter 45 and press tan
Your answer should be 1, if it is not, it may not be in degree mode.
Ex. In the triangle below, find the length of BC.
C
27˚
A
B
5.0 cm
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Ex. Determine the length of VX to the nearest tenth of a centimetre.
X
42˚
V
tan 42˚ =
7.2
𝑋𝑉
7.2 cm
W
In triangle ABC, ∠B = 90˚, and ∠C = 30˚, what is the measure of ∠A?
What is the sum of all angle in a triangle?
Ex. A surveyor wants to determine the width of a river for a proposed
bridge. The distance from the surveyor to the proposed bridge site is
400m. The surveyor uses a theodolite to measure angles. The surveyor
measures a 31˚ angle to the bridge site across the river. What is the width
of the river, to the nearest metre?
tan𝜗 =
Proposed
bridge
31˚
400m
Assignment: pg. 75-77 #3-11, 13, 14, 16-18, 21
Assignment: pg. 82-83 #3-11, 14, 16
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𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
MATH LAB: MEASURING AN INACCESSIBLE HEIGHT
Learning Outcome: Learn to determine a height that cannot be measured
directly.
Key Math Learnings: A clinometers, a measuring tape, and the tangent
ratio can be used to solve a problem that involves indirect measurement.
Tree farmers use a clinometers to measure the angle between a horizontal
line and the line of sight to the top of a tree. They measure the distance to
the base of the tree. How can they then use the tangent ratio to calculate
the height of the tree?
On page 85, read over Part A which describes how to build a clinometers.
Now get with a partner and complete Parts A-J on pages 85-86.
Fill in the chart provided:
Object Measuring
Estimated
Height
Horizontal
distance from
object
Sketch a diagram below (Part G)
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Acute angle
measurement
Distance eye is
from ground
What is the height of the object you measured? Show all work below.
Does the calculation seem reasonable?
Compare your results with your partner. Does the height of your eye affect
the measurement?
Complete Assess your understanding #1-3 on page 86.
If time permits, measure a second object. Fill in the chart on the previous
page for your second object.
Sketch of 2nd object:
Assignment: pg. 86 #1-3
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LESSON 3: THE SINE AND COSINE RATIOS
Learning Outcome: Learn to develop and apply the sine and cosine ratios
to determine angle measures. Learn to use the sine and cosine ratios to
determine lengths indirectly.
Key Math Learnings: In a right triangle, there are two trigonometric ratios
that relate the opposite side to the hypotenuse and the adjacent side to the
hypotenuse. Because the sine and cosine ratios are constant for a given
angle in a right triangle, they can be applied to calculate the length of a side
of any right triangle.
Making Connections:
Has anyone ever travelled on a train?
If not, while travelling by car, have you ever noticed a train going through
tunnels?
Why do you think tunnels were being constructed for trains?
Now if tunnels were not being used, the trains would have to travel along
an incline during transit. Here is a diagram of the track before the tunnels
were constructed.
Point B
6.6 km
297 m
Point A
How would you determine the angle of inclination of the track?
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In a right triangle, the ratios that relate each leg to the hypotenuse depend
only on the measure of the acute angle, and not on the size of the triangle.
These ratios are called the sine ratio and the cosine ratio.
sin 𝐴 =
cos 𝐴 =
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ∠𝐴
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
C
Opposite
∠A
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 ∠𝐴
hypotenuse
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
B
Adjacent to ∠
A
A
The tangent, sine and cosine are called primary trigonometric ratios. The
trigonometry means “three angle measure.”
Ex. Determine sin 32 and cos 32, rounded to 4 decimal places and explain
the meaning of the results. Label the triangle first.
Sin(32) = 0.5299
Cos(32) = 0.8480
32˚
Sin 32 = 0.5299 means that the side opposite
the 32 angle is 0.5299 times as long as the
hypotenuse
cos 32 = 0.8480 means that the side
adjacent to the 32 angle is 0.8480 as
long as the hypotenuse
12
B
Ex. Write each trigonometric ratio
a. sin A
b. cos A
c. sin B
d. cos B
5
4
A
3
sin 𝐴 =
,
cos 𝐴 =
C
sin 𝐵 =
,
cos 𝐵 =
Ex. Determine the measures of ∠K and ∠M to the nearest tenth of a
degree.
8
K
M
3
N
Ex. In the World Cup Downhill held at Panorama Mountain Village in BC,
the skiers raced 3514m down the mountain. If the vertical height of the
course was 984m, determine the average angle of the ski course with the
ground.
3514m
984m
𝜃
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A surveyor can measure an angle precisely using an instrument called a
transit. A measuring tape is used to measure distances.
The diagram shows measurements taken by surveyors. Discuss with a
partner at least two different ways how you could determine the distance
between the transit and the survey pole?
Survey Pole
Survey Stake
110 m
46.5˚
transit
Ex. Determine the length of PQ to the nearest tenth of a centimetre.
P
10.4 cm
Be sure to label the sides first. Notice what
information is given, and what you need to find
out. Use SOH CAH TOA to determine
which ratio to use:
Given hypotenuse side and need to find
opposite side, use sin
67˚
R
Q
14
Ex. In right triangle PQR, R  90, P  24, and PQ = 7.5cm. Calculate the
lengths of RQ and PR to the nearest tenth of a centimetre.
Q
7.5 cm
P
24˚
R
Ex. A pilot starts his takeoff and climbs steadily at an angle of 12.2˚.
Determine the horizontal distance the plane has travelled when it has
climbed 5.4km along its flight path. Express your answer to the nearest
tenth of a kilometre.
5.4 km
12.2˚
x
Assignment: pg. 95-96 #1-13, 15,
Assignment: pg. 101-102 #1-10, 12
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LESSON 5: APPLYING THE TRIGONOMETRIC RATIOS
Learning Outcome: Learn to use a primary trigonometric ratio to solve a
problem modeled by a right triangle.
Key Math Learnings: Any of the trigonometric ratios can be used in a right
triangle, along with the Pythagorean Theorem, if sufficient information
about the triangle is known.
When we calculate the measures of all the angles and all the lengths in a
right triangle, we solve the triangle. We can use any of the three primary
trigonometric ratios to do this.
The basic strategy to solve triangle is:
1. Notice where the acute angle is and label the sides opposite,
adjacent and hypotenuse.
2. Notice what is given and what you need to solve.
3. Decide on a trigonometric ratio that can be used to solve
The following Acronym can be helpful in remembering the ratios:
SOH CAH TOA
sin =
opposite
hypotenuse
cos =
adjacent
hypotenuse
16
tan =
opposite
adjacent
Ex. Solve the following triangle:
C
5 cm
A
B
12 cm
Ex. Solve the following triangle:
A
22cm
42˚
C
B
17
Ex. Solve the following Triangles for the indicated variable:
a.
b.
y
x
32.4 m
18.3 m
83m
x
37 ̊
z
Assignment: pg. 111-112 #3-9, 11-13, 16
Assignment: pg. 88 #1-5, pg. 104 #1-5
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LESSON 6: SOLVING PROBLEMS INVOLVING MORE THAN ONE
RIGHT TRIANGLE
Learning Outcome: Learn to use trigonometry to solve problems modeled
by more than one right triangle.
Key Math Learnings: Many problems involve one or more right triangles
that may be in different planes.
The Muttart Conservatory in Edmonton has four climate-controlled square
pyramids, each representing a different climatic zone. Each of the tropical
and temperate pyramids is 24m high and the side length of its base is 26m.
Work with a partner.
Use this square based pyramid and label its height and base with the
measurements above.
Draw right triangles on the drawing that would help you determine the
angle between the edges of the pyramid at its apex.
How could you use trigonometry to help you determine this angle?
19
A
Need to use
Pythagorean theorem
to solve
24m
E
B
C
13m
D
We can use trigonometry to solve problems that can be modeled using
right triangles. When more than one right triangle is involved, we have to
decide which triangles to start with.
Ex. Calculate the length of CD to the nearest tenth of a centimetre.
C
B
47˚
26˚
A
D
4.2cm
20
Angle of elevation and angle of depression:
Both angles of elevation and angles of depression are always measured
from the horizontal. The angle of elevation looks from the horizontal
upwards:
Angle of elevation
And the angle of depression looks from the horizontal downwards:
Angle of depression
For example:
In Triangle ABC, the angle of elevation and the angle of depression are
indicated.
Angle of Depression
Angle of Elevation
Note: The angle of elevation and the angle of depression are always equal
Ex. From a height of 50m in his fire tower, a ranger observes the
beginnings of two fires. One fire is due west at an angle of depression of
9˚. The other fire is due east at an angle of depression of 7˚. What is the
distance between the two fires?
9˚
7˚
50 m
21
Need to solve for each triangle separately, and recognize that the angle of
elevation and depression are equal.
7˚
9˚
7˚
9˚
Ex. A surveyor stands at a window on the 9th floor of an office tower. He
uses a clinometers to measure the angles of elevation and depression of
the top and the base of a taller building. The surveyor sketches this plan of
his measurements. Determine the height of the taller building to the
nearest tenth of a metre.
31˚
42˚
39m
Assignment: pg. 18-121 #1-9, 11, 13-14, 18, 20
Chapter Quiz – (1 class)
Chapter Review – pg. 124-126 (1 class)
Assignment: pg. 124-126 #1-9, 11-15, 17-20, 22, 23
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