Chapter 4-‐ Trigonometry
Lesson Package
MCR3U
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Table of Contents Lesson 1: Special Angles …………………………………..…………………….…………………pg. 3-‐6 Lesson 2: Special Angles and Obtuse Angles……………………………………………….pg. 7-‐9 Lesson 3: Related and Co-‐terminal Angles………………………………………….……pg. 10-‐13 Lesson 4: Reciprocal Trig Ratios……………………………………………………….…….pg. 14-‐15 Lesson 5: Problems in 2 Dimensions………….……………………………………..……..pg. 16-‐18 Lesson 6: Problems in 3 Dimensions………………………………………………………..pg. 19-‐21 Lesson 7: Ambiguous Case of Sine…………………………………………….……………..pg. 22-‐24 Lesson 8: Trig Identities 1………………………………………………………..……………..pg. 25-‐27 Lesson 9: Trig Identities 2………………………………………………….…….……………..pg. 28-‐29
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Special Angles
Trigonometry Review
- There are three primary trigonometric ratios for right angled triangles.
Sine, Cosine, and Tangent
If we know a right angle triangle has an angle of 'θ ', all other right angle triangles with an angle of 'θ ' are similar and therefore have equivalent ratios of corresponding sides.
sinθ
cosθ
tanθ =
=
= opposite
opposite
hypotenuse
hypotenuse
adjacent
adjacent
Acronym:
S TCO OH H
AA
SOH CAH TOA
1x
a) Finding a missing side
DO IT NOW!
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b) Finding a missing angle
2 The idea of Special Angles:
We look at two types of right triangles
i) isosceles: 45 - 45 - 90 ii) equilateral (split in half): 30 - 60 - 90
We get EXACT values for the three primary trig ratios using these special triangles -
Special Angles
Special Angles1) A triangle
45O
3 primary trig ratios
Special Angles2) A triangle
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Note: this is an equilateral triangle
3 primary trig ratios
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Example 1: Use special triangles to find the exact values of all sides and angles:
4cm
Example 2: Use special triangles to find the exact values of all sides and angles:
Example 3: Determine the exact value of…
Rationalizing the Denominator(gets rid of the square root in the denominator)
a)
b)
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Example 4: Find the exact value of…
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Special Angles and Obtuse AnglesHomework Questions?
Review of Special Angles
Isosceles 45-45-90 Half Equilateral 30-60-90
45O
45O
1
1√2 2
1
√3
Terminal Arm:
The arm of an angle that meets the initial arm at the origin and rotates around the origin counterclockwise to form a positive angle (or clockwise to form a negative angle). It determines what 𝜃 is.
Initial Arm:
First arm of an angle drawn on a Cartesian plane that meets the other (terminal) arm of the angle at the origin. Is always at 0 degrees (does not move).
𝜃
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Any angle over 90 has a reference angle. The reference angle is between 0 and 90 and helps us determine the exact trig ratios when we are given an obtuse angle (angle over 90 degrees). The reference angle is the angle between the terminal arm and the closest x-axis (0/360 or 180).
120
Using Reference Angles to Find Trig Ratios for Obtuse Angles
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When finding the trig ratios of positive angles we are rotating counter clockwise from 0 degrees toward 360. The CAST rule helps us determine which trig ratios are positive in each quadrant
C
AS
T
all
cosine
sine
tan
CAST Rule
Quadrant 1Quadrant 2
Quadrant 3 Quadrant 4
Example 1: Find the reference angle
a) b)
c) d)
Example 2: Find the exact value of
a) b)
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c) d)So far, we have talked about rotating counter clock wise. We can also rotate clock wise from the x-axis but the angles are negative.
Negative Angles
-45
315
-45 and 315 are the same angle.
Example 3: Find the exact values of
a) b)
c)
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4.2 Co terminal and Related Angles
In this section, you will learn how to identify different angles that have the same trigonometric ratio, as well as learn how they are related.
To do this you will have to visualize the terminal arm rotating around a circle centred at the origin of a grid with a radius of r. This is done so that we can extend our understanding of trig functions for a broader class of angles and see how different angles are related.
http://www.mathsisfun.com/geometry/unit-circle.html
𝜃x
yr
The circle being used has radius r. The radius and the coordinates of a point on the circle (x, y) are related to the primary trig ratios. Study the circle and write expressions for sin 𝜃, cos 𝜃, and tan 𝜃 in terms of x, y, and r.
(x, y)
DO IT NOW!
sin 𝜃 =
cos 𝜃 =
tan 𝜃 =
When finding the trig ratios of positive angles we are rotating counter clockwise from 0 degrees toward 360. The CAST rule helps us determine which trig ratios are positive in each quadrant
C
AS
T
all
cosine
sine
tan
CAST Rule
Quadrant 1Quadrant 2
Quadrant 3 Quadrant 4
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Ex. Given that and that lies in the first quadrant, determine exact values for and
Example 1 Example 2
a) Find both angles of 𝜃 for question 1 between 0 and 360 that have the same sine value
b) Find another angle that will have the same cosine ratio as #1
Example 3: Find both angles between 0° and 360° where sin𝜃 = -0.7
Example 4: Find both angles between 0° and 360° where tanθ=2.1
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Example 5: The point (-7, 19) lies on the terminal arm. Find the angle to the terminal arm. Find the reference angle.
Co-terminal Angles: Angles in standard position that have the same terminal arm.
Starting at 30° and rotating 360° counterclockwise will bring you back to the same terminal arm.
30° + 360° = 390°
Therefore 30° and 390° are co-terminal angles.
Note: you can also move clockwise and get negative angles as co-terminal angles.
Example 6: Find three other co-terminal angles of 60°
Homework: 4.2 worksheet
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4.3 Reciprocal Trig Ratios
DO IT NOW!Using a unit circle (circle with a radius of 1), determine the three primary trig ratios for each of the following angles.
(0,1)
(-1,0)
(0,-1)
(1,0)
Remember:
Reciprocal Trig Ratios
Reciprocals: two expressions that have a product of 1.
Review of Special Angles
Isosceles 45-45-90 Half Equilateral 30-60-90
45O
45O
1
1√2 2
1
√3
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Example 1: Use your knowledge of special angles to complete the following chart.
Example 2: Determine the exact value of all six trig ratios for angle A.
A
B
C 5m
12m 13m
𝛳
Example 3: Each angle is in the first quadrant. Determine the measure of each angle.
a) b)
c)
Example 4: Determine two angles between 0 and 360 that have a cosecant of -2
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4.4 Problems in Two Dimensions
Review
Right triangle problems SOH CAH TOA
Oblique triangle problemsSine Law
Cosine Law
Sine law
Used when:
i) two sides and an opposite angle are knownii) two angles and one side are known
Cosine Law
Used When:
- two sides and a containedangle are known
Used When:
- all three sides are known
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Example 1: Jonathan needs a new rope for his flagpole but is unsure of the length required. He measures a distance of 10m away from the base of the pole. From this point, the angle of elevation to the top of the pole is 42°. What is the height of the pole, to the nearest tenth of a metre?
Example 2: Pam, Steven and Rachel are standing on a soccer field. Steven and Rachel are 23m apart. From Steven's point of view, the other two are separated by 72°. From Pam's point of view, the others are separated by an angle of 55°.
a) Sketch a diagram
b) Determine the distance from Pam to Rachel. Example 3: A drive belt wraps around three pulleys as shown. Find the perimeter of the drive belt to the nearest tenth of a cm.
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F
G
H5.1 cm
4.8 cm
6.2 cm
Example 4: Find the measure of angle G
Complete Worksheet
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4.5 Problems in 3-D
Review
Right triangle problems SOH CAH TOA
Oblique triangle problemsSine Law
Cosine Law
Sine law
Used when:
i) two sides and an opposite angle are knownii) two angles and one side are known
Cosine Law
Used When:
- two sides and a containedangle are known
Used When:
- all three sides are known
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Example 1: A radio antenna lies due north of Sam's house. Sam walks to Elena's house, a distance of 1200m, 50° east of north. From Elena's house, the antenna appears due west, with an angle of elevation of 12°. Determine the height of the antenna, to the nearest metre.
Step 1: find the distance from Elena's house to the base of the flag pole.
Step 2: find the height of the antenna.
Example 2: Justine is flying her hot-air balloon. She reports that her position is over a golf course located halfway between Emerytown and Fosterville, at an altitude of 1500m. Fosterville is 16.0km east of Danburg, and Emerytown is 16.5km from Danburg, in a direction 42° south of east. What is the angle of elevation seen from Danburg, to the nearest degree.
Step 1: find the measure of EF
Step 2: find the measure of angle E
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Step 3: find the measure of DG
Step 4: find the angle of elevation seen from Danburg.
Note: EG is half of EF
Example 3: A surveyor is on one side of a river. On the other side is a cliff of unknown height that she wants to measure. The surveyor lays out a baseline AB of length 150,. From point A, she selects pint C at the base of the cliff and measures ∠CAB to be 51°. She selects point D on top of the cliff directly above C and measures an angle of elevation of 32°. She moves to point B and measures ∠CBA as 62°. Determine the height of the cliff to the nearest meter.
Step 1: Find the length of ACStep 2: Find the height of the cliff (side DC)
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The Ambiguous Case of Sine Law
ambiguous case: a problem that has two or more solutions
Sine law
Used when:
i) two sides and an opposite angle are known
ii) two angles and one side are known
You must always consider the ambiguous case of sine when you have an oblique triangle with two sides and an opposite angle given.
Example: The length of side a and side b are given. The measure of angle A is given. Angle B could be an acute or obtuse angle. You must consider both cases.
When to Consider the Ambiguous Case
a b
A
ab
AAcute triangle Obtuse triangle
C
AS
T
all
cosine
sine
tan
Quadrant 1Quadrant 2
Quadrant 3 Quadrant 4
CAST Rule and the Ambiguous Case of Sine
𝜃
It is possible for angles in quadrant 1 and 2 to have equivalent positive sine ratios. Therefore, both scenarios must be considered!
𝜃180 - 𝜃
Scenario 1:
Scenario 2:
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The angle across from the extra side can either be acute or obtuse.
Case 1: Use the sine law as usual Case 2: 180° - answer from Case 1
How to solve for both cases:
How to Solve for the Ambiguous Case of Sine
Example 1: In triangle ABC, side a = 12 cm, side b = 17 cm, and A = 21°. Find the measure of angle B.
Example 2: In triangle ABC, side a = 8 cm, side c = 10 cm, and A = 34°. Find angle C.
Example 3: In triangle ABC, side a = 14 cm, side b = 17 cm, and A = 54°.
a) Find angle B. b) Find angle Cc) Find side c
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Example 4: A lighthouse at point L is 10km from a yacht at point Y and 8 km from a sailboat at point B. From the lighthouse to the yacht to the sailboat is an angle of 48°.
a) Is it necessary to consider the ambiguous case?
b) Sketch the two possible diagrams for the situation.
c) Find the distance from the yacht to the sail boat. If there are two possible answers, determine both.
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4.6 Trigonometric Identitiespart 1
Identity: an equation that is always true, regardless of the value of the variable.
The Fundamental Trig Identitites
Tips for Proving Complex Identities Example 1:
L.S. R.S.
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Example 2:
L.S. R.S.
Example 3: Prove that
L.S. R.S.
Example 4:
L.S. R.S.
Example 5:
L.S. R.S.
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Example 6: Prove that
L.S. R.S.
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4.6 Trigonometric Identitiespart 2
DO IT NOW! Prove:
LS RS
The Fundamental Trig IdentititesTips for Proving Complex Identities
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Example 1: Prove that
Example 2: Prove that
Example 3:
Prove that
Example 4: Prove that
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