Download Math 10C Trigonometry

MATHEMATICS DEPARTMENT
Math 10C Honours
Trigonometry
Teacher: Mr. Trisevic
Name:
1
Unit 3: Trigonometry
Lesson 1: Primary Trigonometric Ratios ....................................................................... 3
Lesson 2: Angle of Elevation (Inclination) and Angle of Depression ........................ 9
Lesson 3: Applications of Right Triangles................................................................... 11
Lesson 4: Angles in Standard Position ........................................................................ 15
Lesson 5: The Sine Law and The Cosine Law ............................................................. 22
2
Lesson 1: Primary Trigonometric Ratios
Outcomes:
1. Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve
problems that involve right triangles.
2. Develop the primary trigonometric ratio and use it to find solutions for angles of
inclination problems.
3. Apply the Pythagorean Theorem.
4. Solve problems that involve ratios.
Trigonometry is the study of triangles, with special attention on the specific study of their sides
and angles. The word trigonometry is a 16th-century Latin derivative from the Greek words for
triangle (trigōnon) and measure (metron).
Throughout the trigonometry unit, we will get to learn about and solve problems involving the
three primary trigonometric functions (𝑠𝑖𝑛, 𝑐𝑜𝑠, 𝑎𝑛𝑑 𝑡𝑎𝑛). Each of these ratios will help us
discover how we can approximate solutions to real-world problems, while also establishing a
strong foundation for the uses of trigonometry beyond Math 10C. From this point forward, we
leave the protractors in the past (of course with occasional use) and learn the power of
trigonometry!
Investigation
A home builder is looking to instal solar panels on all the new homes being built. They are
looking at one side of the roof and decide they will place it to match the pitch of the roof. Which
side represents the pitch? How can we describe the other two sides?
The angle of inclination is the indicated angle. That is, it is the angle that forms an acute angle
with the horizontal line segment of the triangle. This is sometimes referred to as the latitude.
Now suppose we take the same triangle, but we want to label each of the sides such that they
are in formal terms according to the Pythagorean Theorem or trigonometric ratios. Let’s use the
bottom right angle as our reference point.
3
Primary Trigonometric Ratios
As mentioned, there will be three primary trigonometric functions. The three functions may be
represented as Sine, Cosine, and Tangent Ratios and are expressed/defined as the following:
𝑠𝑖𝑛𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑐𝑜𝑠𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑡𝑎𝑛𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
An easy way to remember which sides are involved in what ratio is to use the acronym:
SOH CAH TOA
Ex. 1 Determine the trigonometric ratios for each of the acute angles of the triangles.
How do we find the length of the hypotenuse?
4
Pythagorean Theorem: For a right triangle with two side lengths and a hypotenuse
____________2 + ____________2 = _________________2
What if we know the angle of a specific point and are given one side length? How can we
find a requested missing side length?
Ex. 2 For each of the triangles, find the measure of the indicated side. Round answers to the
nearest tenth.
a)
b) Side length KJ
c)
d) Side length TP
5
The sine, cosine, and tangent functions can be found for any right tringle using the
𝑠𝑖𝑛, 𝑐𝑜𝑠, 𝑎𝑛𝑑 𝑡𝑎𝑛 ratios. We just need to know what the appropriate side lengths and angles are.
Now, what if we would like the find the angle for a specific point of a right triangle?
Finding An Angle Measure
That brings us to exploring the method of solving for angles of the primary trigonometric
functions. This is done by taking the inverse function for each of the trigonometric functions.
𝜃 = 𝑠𝑖𝑛−1 (
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
)
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝜃 = 𝑐𝑜𝑠 −1 (
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
)
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝜃 = 𝑡𝑎𝑛−1 (
)
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
These inverse functions can be accessed on your calculator by pressing 2𝑛𝑑 →
sin 𝑜𝑟 cos 𝑜𝑟 𝑡𝑎𝑛 → 𝑣𝑎𝑙𝑢𝑒 𝑏𝑒𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑒𝑑 → 𝑒𝑛𝑡𝑒𝑟.
If we want to calculate these angles using our calculator, we need to change our calculator
settings so that they appropriately match the units we are calculating.
Steps To Change Mode on Calculator
1. ‘MODE’ button
2. Scroll down to ‘RADIAN DEGREE’
3. Use the right cursor to move to ‘DEGREE’ and press ‘ENTER’
You will not need to use radian mode until grade 11 and 12. Until, then keep your calculator in
degree mode!
Ex. 3 Determine the measure of each angle for the following trigonometric functions. Round
answers to the nearest tenth.
a) 𝑠𝑖𝑛𝜃 = 0.25
c) 𝑡𝑎𝑛𝜃 = 4.3
b) 𝑐𝑜𝑠𝜃 = 0.64
d) 𝑠𝑖𝑛𝜃 = 1.8487
6
Properties and Theorems of Angles
1. Triangle Sum Theorem: The sum, sigma (∑), of the inner angles of a triangle is equal
to 180°.
2. Complimentary Angles: Two angles with a sum (∑) of 90°. Supplementary Angles:
Two angles with a sum (∑) of 180°.
3. Acute Angle: Angle between 0° and 90°. Obtuse Angle: Angle between 90°
and 180°. Right Angle: Angle that is 90°.
Now let’s put the ideas established to work and solve triangles that have unknown angles.
Ex. 4 Determine the measure of the indicated angles for the following triangles. Make sure
answers are rounded to the nearest tenth.
a) ∠𝐺 and ∠𝐻
b) ∠𝐾 and ∠𝑀
7
Finally, what if we want to solve for both missing angles and side lengths? If we are solving for
both for a given right triangle, we say that this is solving the triangle.
Ex. 5 Solve the following triangles. Round each answer to the nearest tenth.
a)
Q
s
r
R
4.5 cm
S
b)
A
5.5 cm
C
B
Ex. 6 In right △DEF, ∠E= 90°, ∠F = 63°, and DF = 7.8 𝑐𝑚. Solve this triangle. State the
measures to the nearest tenth.
8
Lesson 2: Angle of Elevation (Inclination) and Angle of Depression
Outcomes:
1. Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve
problems that involve right triangles.
2. Define and explore the connections between angles of elevation and depression.
3. Solve problems that involve ratios.
Angles can be classified into two types when dealing with right triangles and real-life situations.
Angle of Elevation: the angle that is formed by the horizontal and the line of sight above the
horizontal.
Angle of Depression: the angle that is formed by the horizontal and the line of sight below the
horizontal.
Now what can we say about the connection between the angle of elevation and the angle of
depression? Let’s do a quick investigation.
Investigation
Suppose a human is on the ground looking up at a bird on the top of a lamp post. The human is
1.7m tall and he is standing 19m from the lamp post. From the humans eyeline, the post is
37.289m tall. If the human and the bird make eye contact, what is the angle each must look up
or down to do so? Round to the nearest whole degree.
9
Thus, the angle of elevation (inclination) and the angle of depression are _________________.
Ex. 1 Natalie is rock climbing and Aaron is belaying (supporting by keeping tension on the
rope). When Aaron pulls the rope taut to the ground, the angle of depression is 73°. If Aaron is
standing 8 ft from the wall, what length of rope is off the ground?
Ex. 2 At night, it is possible to make precise measurements of cloud height using a search light.
An alidade (surveying/measurement tool) is set 720 ft away from the search light (where the
search light is directly below the cloud). It measures the angle of elevation to the place where
the light strikes the cloud to be 35°. What is the altitude of the cloud? Express your answer to
the nearest foot.
Ex. 3 An airplane is observed by an air traffic controller at an angle of elevation of 52°. The
airplane is 850m above the observation deck of the tower. What is the distance from the
airplane to the tower? Express your answer to the nearest meter.
10
Lesson 3: Applications of Right Triangles
Outcomes:
1. Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve
problems that involve right triangles.
2. Solve problems that involve multiple right triangles.
3. Summarize solutions in proper sentence structure with mathematical notation.
4. Solve problems that involve ratios.
During this lesson, it will be important to remember that when solving for solutions, do not
round until the final solution. Try to keep as many decimal places as possible or have the
values ready to use in your calculator. This will help lead to more accurate solutions.
When completing problems and showing work, it is fine to write numbers with decimals as
56.873 …
Then round the number as asked to in the specific given question.
Ex. 1 From a height of 50 m in his fire tower near Francois Lake, BC, 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, to the
nearest metre?
11
Ex. 2 Calculate the length of BA to the nearest tenth of a centimetre.
Ex. 3 Determine the measure of ∠CAB to the nearest degree.
12
Ex. 4 From the top of a 20m building, a surveyor measured the angle of elevation of the top of
another building and the angle of depression of the base of that building. The Surveyor
sketched this plan of her measurements as shown below. Determine the height of the taller
building to the nearest tenth of a meter.
Ex. 5 A communications tower is 35 m tall. From a point due north of the tower, Tannis
measures the angle of elevation of the top of the tower as 70°. Her brother Leif, who is due east
of the tower, measures the angle of elevation of the top of the tower as 50°. How far apart are
the students to the nearest metre? The diagram is not drawn to scale.
13
Ex. 6 A student wanted to know the distance between two particular carvings on a spirit pole.
She measured the angle of elevation of each carving 15.0 m from the base of the pole. The
student drew the sketch below. What is the distance between the carvings to the nearest tenth
of a metre?
14
Lesson 4: Angles in Standard Position
Outcomes:
1. Develop an understanding of angles in standard position [0° to 360°].
2. Solve problems, using the three primary trigonometric ratios for angles in standard
position [0° to 360°].
3. Use exact values to describe solutions of sin, cos, and tan for angles in standard
position.
4. Classify the quadrant where angles in standard position terminate.
5. Solve problems that involve ratios.
In this lesson, we are going to be exploring a grade 11 mathematics topic. In this lesson, we
develop an understanding of angles in standard position using the Cartesian plane. A first part
of developing this understanding will be introducing the notation and terminology that we will be
using.
First, an Angle in Standard Position is an angle that has an initial arm on the positive 𝑥-axis
and has its vertex at the origin.
Now suppose the Cartesian plane is drawn as follows:
Initial Arm: Arm of an angle in standard position that lies on the 𝑥-axis and begins at the origin.
Terminal Arm: Arm of an angle in standard position that meets the initial arm at the origin to
form an angle.
Direction of rotation maters! When the terminal arm rotates to create a specific angle, the
direction of ration determines whether the angle is positive or negative.
•
•
When the terminal arm rotates counter-clockwise to create an angle in standard
position, we get a positive angle.
When the terminal arm rotates clockwise to create an angle in standard position, we
get a negative angle.
15
Properties of the Cartesian Plane and Angles in Standard Position
Given the blank Cartesian plane, let’s label the plane with the degrees and quadrant
numbers.
𝑦
𝑥
As long as we have an angle in standard position that lies on the Cartesian plane from [0° to
360°], then we have a principal angle. That is, an angle that is _______________________.
Ex. 1 Sketch the following angles in standard position, listing the quadrant the angle terminates
in.
a) 20°
b) 300°
c) 220°
d) 96°
Ex. 2 Sketch the following angles in standard position, then state the principal angle.
a) -16°
b) -90°
c) -190°
d) -320°
16
In the previous example, we took negative angles and related them to positive ones (principal
angle). This was relating two numbers that described the same location on the Cartesian Plane.
There is a specific way we classify this:
Co-Terminal Angles are angles in standard position that are in the same position as the
principal angle after more than one full rotation (positive or negative direction).
Ex. 3 Suppose an angle in standard position of 120° is given. Find one positive and one
negative angle that are co-terminal to this angle.
𝑦
𝑥
We may also describe angles with reference angles: the angle that is created between a
terminal arm and the 𝑥-axis. The reference angle for an angle in standard position may be
denoted by 𝜃𝑅 , where 0° ≤ 𝜃𝑅 ≤ 90°. The reference angle will always be positive and have a
value from 0° to 90°.
Ex. 4 Sketch each of the following angles in standard position, then indicate the value of the
reference angle.
a) 50°
b) 340°
c) -220°
d) -160°
17
Ex. 5 If the reference angle, 𝜃𝑅 , is 50° given, draw and list all the possible angles from 0° ≤ 𝜃 ≤
360° described by this reference angle.
𝑦
𝑥
Summary
Quadrant
Value of Angle in
Standard Position,
𝜽
I
_____° ≤ 𝜃 ≤ _____°
II
_____° ≤ 𝜃 ≤ _____°
III
_____° ≤ 𝜃 ≤ _____°
IV
_____° ≤ 𝜃 ≤ _____°
Sketch
Reference Angle, 𝜽𝑹
18
Now let’s apply and extend the ideas we have just learned. Below is some angle in standard
position, 𝜃, that has a full circle draw around it. At the edge, or circumference, there is a point
(𝑥, 𝑦).
What is the measure 𝑟 represent in this situation?
According to the theorem from measurement, ___________________________________, how
can we calculate the value of 𝑟?
If we write our three primary trigonometric ratios in terms of 𝑥, 𝑦 and 𝑟 we get
𝑠𝑖𝑛𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑐𝑜𝑠𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑡𝑎𝑛𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Or
𝑠𝑖𝑛𝜃 =
𝑐𝑜𝑠𝜃 =
𝑡𝑎𝑛𝜃 =
19
Ex. 6 The point (−8, 15) lies on the terminal arm of an angle 𝜃 in standard position. Determine
the exact rations for 𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃, and 𝑡𝑎𝑛𝜃, and then find the value of 𝜃 to the nearest degree.
The CAST Rule
The CAST rule is an important rule when finding the exact values and angles of trigonometric
ratios in standard position. The CAST rule is an easy way to remember which quadrant of the
Cartesian plane each of the trigonometric ratios are positive.
The classification starts at quadrant IV and each letter is assigned counter-clockwise. Let’s
assemble this below.
𝑦
𝑥
Note: CAST stands for
Cos, All, Sin, Tan. That is,
the ratio is positive
where C, A, S, and T are
assigned.
The CAST rule may help us find missing trigonometric ratios or angles in standard position.
Let’s see how the rule is a guide in our approach to solving problems.
20
3
Ex. 7 Suppose there is an angle in standard position 𝑐𝑜𝑠𝜃 = 4 , where 𝑠𝑖𝑛𝜃 > 0. Determine the
exact values of 𝑠𝑖𝑛𝜃 and 𝑡𝑎𝑛𝜃. Also, determine the measure of 𝜃 to the nearest degree.
Ex. 8 Suppose there is an angle in standard position 𝑠𝑖𝑛𝜃 = −
√3
,
2
where 𝑡𝑎𝑛𝜃 > 0. Determine
exact values of 𝑐𝑜𝑠𝜃 and 𝑡𝑎𝑛𝜃. Also, determine the measure of 𝜃 to the nearest degree.
21
Lesson 5: The Sine Law and The Cosine Law
Outcomes:
1. Solve problems, using the sine law and cosine law.
2. Recognize when to use the sin law, or use the cosine law to solve a problem by the
information given.
3. Draw diagrams to represent data given.
4. Solve problems that involve ratios.
We have been solving triangle’s using right-triangle trigonometry. That is, we have been solving
problems that include right-angles triangles. Now, let’s look at problems that include oblique
triangles (triangles without a right-angle).
Part I: The Sine Law
Investigation
Suppose we have the following oblique triangle
a) If we label the triangle according to the vertices and side lengths, then each angle
should be opposite of its designated side length, and vice versa.
b) Suppose we now draw in the height of the triangle. Where does this belong? Where do
we draw the line?
c) Consider the lower two vertices. If we represent each of the 𝑆𝑖𝑛 ratios at these points,
what do they equal (in terms of variables)?
22
d) Solve for the height. Let’s compare our two 𝑆𝑖𝑛 ratios. What is the outcome?
To summarize our findings into a general rule, then we have that:
The Sine Law is the relationship between the sides and the angles of a triangle. Suppose we
have a triangle △ABC where 𝑎, 𝑏, and 𝑐 are the sides of the triangles opposite to the angles
∠A, ∠B, and ∠C respectively. Then
𝑎
𝑏
𝑐
=
=
𝑆𝑖𝑛 𝐴 𝑆𝑖𝑛 𝐵 𝑆𝑖𝑛 𝐶
Or
𝑆𝑖𝑛 𝐴 𝑆𝑖𝑛 𝐵 𝑆𝑖𝑛 𝐶
=
=
𝑎
𝑏
𝑐
Ex. 1 Solve for the unknown side or angle in each of the following.
a)
𝑎
sin 35°
= sin 40°
10
b)
𝑏
sin 48°
= sin 75°
65
c)
sin 𝜃
12
=
sin 50°
65
d)
sin 𝐵
25
=
sin 62 °
32
23
Ex. 2 Determine the length of AB for each of the following.
a)
b)
Ex. 3 Determine the measure of the indicated angle for each of the following.
a)
b)
Ex. 4 In △ABC, ∠A = 50°, ∠B = 50°, and AC = 27 m. Find the length of AB.
24
Ex. 5 Solve each of the following triangles (find all measures and side lengths).
a)
b)
Part II: The Cosine Law
Investigation
Suppose we begin with the same oblique triangle that we began with for the section on The
Sine Law.
1. Let’s label the triangle as we have done before. Label each of the vertices and side
lengths. Label the height of the triangle as well.
2. Suppose we want to find the side length on the right-hand side of the triangle. Let’s use
the labels for our sides and angles to form a solution.
25
To summarize what we have found, we can say that
The Cosine Law describes the relationship of the cosine of an angle and each of the side
lengths of a triangle. That is,
For some triangle △ABC where 𝑎, 𝑏, and 𝑐 are the sides of the triangles opposite to the
angles ∠A, ∠B, and ∠C respectively. Then
𝑐 2 = 𝑎 2 + 𝑏2 − 2𝑎𝑏𝐶𝑜𝑠C
And similarly,
𝑎 2 = 𝑏2 + 𝑐 2 − 2𝑏𝑐𝐶𝑜𝑠A
𝑏2 = 𝑎 2 + 𝑐 2 − 2𝑎𝑐𝐶𝑜𝑠B
When relating the cosine of an angle to the three side lengths of a triangle, what pattern do we
notice within The Cosine Law?
Ex. 1 In △ABC, a = 11, b = 5, and ∠C = 20°. Sketch a diagram and determine the length of the
unknown side and the measures of the unknown angles, to the nearest tenth.
26
Ex. 2 Find the length of the third side of each triangle.
a)
b)
Ex. 3 Determine the measure of the following indicated angles.
a)
b)
27