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Trigonometry and the Unit Circle
Chapter 4
Topics• Demonstrate an understanding of angles in
standard position, expressed in degrees and radians.
• Develop and apply the equation of the unit circle.
• Solve problems, using the six trigonometric ratios for angles expressed in radians and degrees.
• Solve, algebraically and graphically, first and second degree trigonometric equations with the domain expressed in degrees and radians.
Special Angles
• Without a calculator determine the following:
A) Sin 30o B) tan 45o C) sin 60o
Special Triangles• 45o-45o-90o
Special Triangles• 30o-60o-90o
From geometry• An angle consists of 2 rays that originate
from a common point called the vertex.
90 180o o vertex
Reflex angle > 180o
Obtuse angle is between 90o and 180o
Acute angle is less than 90o
0 90o o
180o
It’s Greek To Me!It is customary to use small letters in the Greek alphabet to symbolize
angle measurement.
alpha beta gamma
theta
phidelta
• To place an angle in standard position, one ray is placed on the postive x-axis with the vertex at the origin. – This side is called the initial side.
– The other side is called the terminal side (or arm).
Initial side
Angle Measurements
• We will be using two different units of measure when talking about angles: Degrees and Radians
• Let’s talk about degrees first. – This was covered in Math 2200
• One degree (1º) is equivalent to a rotation of of one revolution. 1
360
Example:
1. Place the following in standard position:
A) 90o B) 180o C) 270o D) 360o
Note: These are called quadrantal angles
These are angles that lie along the x-axis or y-axis
o1 of 360
4o1
of 3602
o3 of 360
4
o360
(1 revolution)
E) 54o F) 120o G) 245o H)-150o
Note:
• If the angle is positive the terminal arm is swept out from the initial side in a counter-clockwise fashion.
• A negative angle is swept out in a clockwise fashion
• It might be helpful to think of a positive rotation as opening upward from standard position, whereas a negative angle opens downward.
• Angles are often classified according to the
quadrant in which their terminal sides lie.
Example:
Name the quadrant in which each angle lies.
50º
208º II I
-75º III IV
Quadrant 1
Quadrant 3
Quadrant 4
Classifying Angles
A)135o B) 220o C) -45o D)-150o
Definition: Reference Angle: This is the acute angle that is formed by the terminal arm of the angle and either the positive or negative x-axis.
Example:Draw the angles in standard position and find the reference angles
R
___R ___R ___R ___R
E) 604o F) 270o G) -345o H)-550o
Note:
• For angles larger than 360o or smaller than -360o we subtract or add multiples of 360o to determine where the angle is.
___R ___R ___R ___R
Exit Card: Sketch each angle in standard position and find the reference angle.
A) 60o B) -125o C)-217o D) 750o
___R ___R ___R ___R
E)-330o F) 242o G) 270o h) 1090o
___R ___R ___R ___R
Radian Measure
• A second way to measure angles is in radians.
So, what is a radian and why
do we use them?
(Why not just use degrees?)
What is a radian?
When the length of the arc
equals the length of the radiusThe angle has a
measure of 1 radian
A radian is an angle measurement that gives the ratio:
length of the arc
length of the radius
= 1 radian
the length of the radius
The angle has a
measure of 2 radians
When the length of the arc
is twice as long as
= 2 radians
the length of the radiusThe angle has a
measure of 3 radians
When the length of the arc
is three times as long as
= 3 radians
What is a radian?
the length of the radiusThe angle has a
measure of π radians
A radian is an angle measurement that gives the ratio:
length of the arc
length of the radius
When the length of the arc
is π times as long as
= π radians
What is a radian?A radian is an angle measurement that gives the ratio:
length of the arc
length of the radius
Radius= r
s
r
Arclength = s
s
r
Why do we use radians?
Radians are very important in Calculus.
The area between y = sin (x) and the x-axis
from 0 < x < 180 is approximately 114.6 when
graphed in degrees.
Using radians:
The area between y = sin (x) and the x-axis
from 0 < x < π is exactly 2 when graphed in
radians.
0
sinlim 1
Finding the limits of trigonometric function and subsequently finding derivatives of trig functions work best when using radians
In radians: In degrees:
0
sinlim 0.01745...
Radian Measure
2 radians corresponds to
radians corresponds to
radians corresponds to2
360
180
90
2 6.28
3.14
1.572
Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–7
Section 4.1, Figure 4.6, Illustration of
Six Radian Lengths, pg. 249
Common Radian Measure
Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–8
Section 4.1, Figure 4.7, Common
Radian Angles, pg. 249
Conversion of angle measurement
• To convert degrees to radians:
• Example 1: Convert the following degree measures into radian measures
A) 45o B) 120o C) 90o
DegreeMeasure 180o
Radian
Measure
180 º = radians
Ex2. Convert the degrees to radian measure.
a) 60 b) 30 c) -54
d) -118 e) 225
Conversion of angle measurement
• To convert radians to degrees:
• Example 3: Convert the following radian measures into degree measures
A) 2 B) C) D)
RadianMeasure
180o
DegreeMeasure
2
3
6
5
4
11
Ex 4. Convert the radians to degrees.
a) b) c)
d) e)
6
11
18
9
12
7𝜋
3
0°
360 °
30 °
45 °
60 °
330 °
315 °
300 °
120 °
135 °
150 °
240 °
225 °
210 °
180 °
90 °
270 °
Degree and Radian Form of “Special”
Angles
34
Co-terminal Angles
Angles that have the same initial and terminal sides are co-terminal.
Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–5
Section 4.1, Figure 4.4, Coterminal
Angles, pg. 248
Angles and are co-terminal.
Finding Co-terminal Angles
You can find an angle that is co-terminal to a given angle by
adding or subtracting multiples of 360º or
Ex 1:
Find one positive and one negative angle that are co-terminal
to 112º. Give exact answers in reduced form.
For a positive co-terminal angle, add 360º : 112º + 360º = 472º
For a negative co-terminal angle, subtract 360º: 112º - 360º = -248º
2
Ex 2: Find one positive and one negative angle that are co-terminal to 700o. Give exact answers in reduced form.
Ex 3. Find one positive and one negative angle that is co-terminal with the angle = in standard position. Give exact answers in reduced form.
Ex 4. Find one positive and one negative angle that is co-terminal with the angle = 3 in standard position. Give approximate answers to 2 decimal places
𝜋
3
For a positive co-terminal angle, add
For a negative co-terminal angle, subtract
2
2
Ex 5. Find all of the angles that are co-terminal with the
angle = from 0 to 6. Give exact answers in reduced
form.
7
5
Ex 6. Find all of the angles that are co-terminal with the angle = 3.2
Radian Measure, Arc Length, and Area
Arc length s of a circle is found with the following
formula:
arc length radius measure of angle
IMPORTANT: ANGLE
MEASURE MUST BE IN
RADIANS TO USE FORMULA!s = r
Recall the formula for radians: s
r
3
= 0.52
arc length to find is in red
s = r
Ex 1. Find the arc length if we have a circle with
a radius of 3 meters and central angle of 0.52
radian.
Ex 2. Find the radius of a
circle in which an arc of 3 km
subtends a central angle of
20°.
Remember: If the measure of the angle is in degrees, we
can't use the formula until we convert it to radians.
s r s
r 0
20180
o
3
9
kmr
27
km
9
Ex 3. Given an arc length of 20 cm cut on a circle of radius 5.4 cm, determine the measure of the central angle in radians and degrees.
Fun One!• During a family vacation, you go to dinner at
the Seattle Space Needle. There is a rotating restaurant at the top of the needle that is circular and has a radius of 40 feet. It makes one rotation per hour.
• At 6:42 p.m., you take a seat at a window table. You finish dinner at 8:28 p.m.
• Through what angle did your position rotate during your stay? How many feet did your position revolve?
• Text: Page 175-179
• # 1, 2, 4, 6, 7, 8, 9,
11 a) c) e) g) , 13,
15c) If the bike tire is 700mm in diameter, how fast is the bike travelling?
Section 4.2Your Friend
THE UNIT CIRCLE
What does “unit circle” really mean?
• It’s a circle with a radius of 1 unitcentred at the origin, (0, 0).
• To find the equation of the unit circle we use the Pythagorean Theorem.
2 2 2a b c
• Point P represents any point on the unit circle.
• Applying the Pythagorean Theorem results in:
• This is the equation of the unit circle.
P(x, y)1
x
y
2 2 21x y
2 2 1x y
• How would the equation differ if the radius was r instead of 1?
• Thus we can generalize the equation of a circle with centre (0,0) and radius r to be x2 + y2 = r2.
Example. Determine the equation centred at the origin satisfy the following
conditions.
A) Radius = 6 B) Radius = 5 C) Radius =
D) Diameter = 1 E) Passes through point (5, -12)
2
Using the unit circle, you should be able to complete the following tasks.
• Given an angle θ in standard position, expressed in degrees or radians, determine the coordinates of the corresponding point on the unit circle.
• Conversely, determine an angle in standard position that corresponds to a given point on the unit circle.
• We start performing these tasks using special angles and then move on to any angles.
• Consider the following unit circle, what are the coordinates for the quadrantal angles?
Place these values on the blank unit circle
30o
60o
45o
EX.
4
Exact ValuesReference Angles of 45o or
, 180 0, 0
2, 360
3
2
2
4
2
2
2
2
1
45
2
2,
2
2
What are the coordinates?
4
Now, reflect the triangle to the second quadrant…
Exact ValuesReference Angles of 45o or
Reference Angles of 30o or 6
30
1 1
2
3
2
6
3
2,1
2
Now, reflect the triangle to the second quadrant.
Reference Angles of 30o or 6
What are the coordinates?
Let’s look at another “family”
Reference Angles of 60o or3
Let’s look at another “family”
Reference Angles of 60o or3
1
, 180 0, 0
2, 360
3
2
2
60
3
2
1
2
1
2,
3
2
3
Now, reflect the triangle to the second quadrant
What are the coordinates?
Examples
1. Determine the coordinates of the corresponding points on the unit circle given the following angles in standard position:
A) 135o B) -210o C) 13
6
2. Determine the radian measure for all angles in standard position on the unit circle given the following corresponding points:
A) (0, 1) B) C) 1 3
,2 2
2 2,
2 2
3. Determine the degree measure for the smallest positive angle in standard position (Principal Angle) on the unit circle given the following corresponding points:
A) (0, -1) B) 1 1,
2 2
4. Determine the coordinates for all points on the unit circle that satisfy the conditions given. Draw a diagram in each case.
A) The x-coordinate is 4
5
4. Determine the coordinates for all points on the unit circle that satisfy the conditions given. Draw a diagram in each case.
B) The y-coordinate is −7
25
5. Determine if the points are on the unit circle. If they are not on the unit circle what is the required radius for the points to be on a circle centred at origin?
A) 2,3 B)2 6
7,−5
7C)
1
4,3
4
6. If P() is a point at the intersection of the terminal arm of angle and the unit circle, determine the exact coordinates of each of the following.
A) P (−𝜋) B) P( 5𝜋
6) C) P(1080o)
• Text Page 186-189
• #1, 2, 3, 4, 5, 11, 15, 16
4.3 Trigonometric Ratios
• In Mathematics 1201 and 2200, you worked with the three primary trigonometric ratios.
• What are they?
• In this section you will be introduced to the reciprocal ratios:
• csc θ, sec θ and cot θ.
• Also, we explore how we can find trigonometric ratios for angles bigger than 90o and for negative angles.– This will be done by applying the trig
ratios to the unit circle.
yr = 1
x
sin1
y
y
cos 1
x
x
sintan
cos
y
x
The Unit Circle
,x y
Terminal Points• A terminal point is the point where the terminal side
of the angle intersects the unit circle.
• Coordinates are (x, y) or (cos , sin )
• Sine fn:
• Cosine fn
• Tangent fn
• Cotangent fn
• Secant fn
• Cosecant fn
sin1
y
P(x, y)
1
x
y
cos1
x
tanyx
cotxy
1sec
x
1csc
y
Note: x2 + y2 = 12
Point P (x, y) can be written as P (cos, sin)
Reciprocal Ratios
• Sine fn:
• Cosine fn
• Tangent fn
• Cotangent
• Secant fn
• Cosecant fn
sin1
y
P(x, y)
1
x
y cos1
x
tanyx
cotxy
1sec
x
1csc
y
Which pairs are reciprocals of each other?
1tan
cot
1cot
tan
1cos
sec
1sec
cos
1sin
csc
1csc
sin
Note:• If
and
then coscot
sin
sintan
cos
y
x
1cot
tan
The Unit Circle
tan0 cot 0
sin0 csc0
cos0 sec0
The Unit Circle
tan cot2 2
sin csc2 2
cos sec2 2
The Unit Circle
tan cot
sin csc
cos sec
Examples1.Evaluate the six trigonometric functions at each real number.
3
2
2
3,
2
1
3
2Sin
3
2Cos
3
2Tan
= y
= x
y
x
2
3
2
1
3 1
2 2
3 2
2 1
3
Where is the terminal arm? What is the reference angle?
What are the coordinates?
2
3,
2
1
3
2Sin
3
2Cos
3
2Tan
2
3
2
1
3
3
2Csc
2
33
3
3
2Cot
3
32
3
2Sec
1
3
3
3
3
3
2
2. Evaluate the six trigonometric functions at each real number.
4
7
2 2,
2 2
Sin
4
7
4
7Cos
4
7Tan
4
7Csc
4
7Sec
4
7Cot
2
2
2
2
-1-1
2
2
So, you think you got it now?
What is the reference angle? What are the coordinates?
Where is the terminal arm?
3. Evaluate the six trigonometric functions at each real number.
(0, -1)2
2Sin
2Cos
2Tan
= y
= x
= -1
= 0
y
x
1
0
DNE
Does Not
Exist
2Sec
1
0
DNE
2Cot
0
1
2Csc
= -1
= 0
• 4.Evaluate the six trigonometric functions at .
• 0660
Page 201 #1
Quiz tomorrowNO CalculatorSome questions MAYcome from homework
For Each Question you must do the following:•What is the exact value for each trigonometric ratio? •Show all necessary work in places provided.
•(Location of terminal arm, reference angle and coordinates of points)
Approximate Values of Trigonometric Ratios
(Using a Calculator)
• Calculators can be use to obtain approximate values for sine, cosine and tangent.
• Most calculators can determine trig values for angles measured in degrees (Deg), and radians (Rad), and even in gradients (Grad)
GRADIENTS (GRADES)
100%
h = 100%
d
risegrad
run
Determine the following
A) Sin 30 B) Sin 30o
In which quadrant does an
angle of 30 terminate?
Using Calculator for finding Exact Values
• Some students rely on their calculator to find exact values.
• For example, if a calculation results in 0.7071, these students have memorized that 0.7071 = ___
If you want to be one of these students you need to know these approximations
1____
2
2____ 2 ____ 3 ____
2
3 2 3 3____ ____ ____
2 3 3
Remember this only helps if you have access to a scientific calculator.
• You can find the values of reciprocal trig functions (csc, sec, cot) using the correct reciprocal relationship.
• Determine the following, correct to 4 decimal places:
A) B) C)sec60 cot( 60)17
csc6
D) sec25°48'30"
If the angle is not exactly to the next degree it can be expressed
as a decimal (most common in math) or in degrees, minutes and
seconds (common in surveying and some navigation).
1 degree = 60 minutes 1 minute = 60 seconds
= 25°48'30" degrees
minutes
seconds
To convert to decimal form use conversion fractions. These are fractions
where the numerator = denominator but in two different units. Put unit on
top you want to convert to and put unit on bottom you want to get rid of.
Let's convert the
seconds to minutes
30"
"60
'1 = 0.5'
1 degree = 60 minutes 1 minute = 60 seconds
= 25°48'30"
Now let's use another conversion fraction to get rid of minutes.
48.5'
'60
1 = .808°
= 25°48.5' = 25.808°
D) sec25°48'30"
sec25.808°
1
cos 25.808°
Page 201#2
Simplify Trigonometric Expressions
• From Mathematics 2200, you should be familiar with performing operations on rational expressions and expressions involving radicals
• Example: Simplify2
2
16
4
xx x
520
2
Simplify Trigonometric Expressions
• In this section the rational expressions will involve exact values of trigonometric functions
• After you obtain an answer you could use a calculator to check your solution.
• The emphasis here, however, is on finding exact values using the unit circle, reference triangles, and mental math strategies.
o
cos sin6
tan 30
Examples
1. 2. 2 2sin cos
6 6
6
7cos
4
7sin
4
3cos
3.
3
4sin
4
3tancos
6sin 2
4.o o
o o
sin45 sin210
sin120 cos30
5.
o
cos sin6tan30
PRACTICE:Find the exact value of the following
expressions:
22 11( ) sin cos
3 6i
11( ) csc cot
3 4ii
2 7( )cot
6iii
o
11cot cos
3 3( )
csc 240iv
Text Page 202. #9
22 11( ) sin cos
3 6i
11( ) csc cot
3 4ii
2 7( )cot
6iii
o
11cot cos
3 3( )
csc 240iv
Finding angles
• Determine, with or without technology, the measures, in degrees or radians, of the angles in a specified domain, – given the value of a trigonometric ratio.
– given a point on the terminal arm of an angle in standard position.
Examples of finding angles given the value of a trigonometric ratio.
1. Determine the value of θ when for the domain −2π ≤ θ ≤ 2π.
• Solution: Determine the reference angle,
• You could think about a triangle with hypotenuse and adjacent side 1.
• Alternatively, you could apply the reciprocal
ratio
sec 2
1cos
2
2
• Once the reference angle is determined, identify the quadrants where the secant ratio is negative.
• The final step focuses on identifying all possible values within the given domain: −2π ≤ θ ≤ 2π.
5 3 3 5, , ,
4 4 4 4
2.Determine the value of θ when for the domain 0≤ θ ≤ 2π.
3sin
2
3.Determine the value of θ when for the domain 0o≤ θ ≤ 360o
csc 2
4.Determine the value of θ when for the domain -2π ≤ θ ≤ 4π
2 3sec
3
Other Examples
5. If the terminal arm of is in the second quadrant and , determine
2
2sin
22 cos2sin
6. If the point is on the terminal arm of , determine
A) The quadrant is in
B) The principal angle for , in radiansRecall: The principal angle is the first positive angle that ends on
the terminal arm
C) tan
2
3,
2
1
7. The point P(-0.7880, 0.6157) is the image of the point (1, 0) rotated through . Find .
Find the reference angle first.
Solve cos R = 0.7880 Take the positive value
R = 38o
Since is in the 2nd quadrant = 180o-38o
= 142o
New Definitions of the Trigonometric Functions
• Consider a circle with radius r, centre at the origin. The terminal side of an angle , in standard position intersects the circle at the point P, with coordinates (x, y).
• Sine fn:
• Cosine fn
• Tangent fn
• Cotangent
• Secant fn
• Cosecant fn
sinyr
P(x, y)
r
x
y
cosxr
tanyx
cotxy
secrx
cscry
Note: x2 + y2 = r2
Point P (x, y) can be written as P (rcos, rsin)
Note:
• The radius is always positive, but the x and y may be positive or negative depending on what quadrant point P lies in.
Ex: Where would be if both x and y are negative?
Examples
1. Find the exact value of csc if
and is in the third quadrant.
Bonus: What is in degrees?
3cos
8
Examples
2. Find the exact value of sin if
and is in the fourth quadrant.
Bonus: What is in radians
5tan
12
Example 3: The following points are on the terminal arm of which intersect a circle with centre (0, 0) and radius r. For each point:
A) Draw a diagram showing as a principal angle, in standard position.
B) Find the radius of the circle.
C) Find the exact values of the 6 trig functions
D) Determine the reference angle R
E) Find
(i) P(3, 4)
(ii) P(-3, 4)
(iii) P(-3, -5)
(iv) P(1, -1)
(v) P(0, -2)
Example 4. The point P(-3.7157, -3.3457) is the point of intersection between the terminal arm of and a circle of radius r centred at the origin.
• A) Find r x2 + y2 = r
= 5• B) Find .
– Find the reference angle.
2 2( 3.7157) ( 3.3457)r
3.7157cos
5R
xr
R = 42o
= 180o + 42o = 222o
5. A point on a circle with radius 8 rotates at 14.125 revolutions per minute. Find the exact location of the point after 3 minutes, assuming that the dot started at (8, 0).
6. Find the approximate measure of all angles when in the domain - ≤ θ ≤ 2. Give answers to 2 decimal places.
sec 3.5
• Page 202 – 205
• #3 – 8, 10-12, 14, 19
• and C4
Introduction to Solving Trigonometric
EquationsSection 4.4
• In Mathematics 2200, you solved simple trigonometric equations of the form
sin 0.5 tan 3.2x x
In Math 3200:• This will now be extended to include
trigonometric equations with all six trigonometric ratios.
• We will solve first and second degree trigonometric equations with the domain expressed in degrees and radians.
First Degree Equations• When solving first degree equations,
rearrangement will sometimes be necessary to isolate the trigonometric ratio.
• Example: 1. Solve 2cos 1 0, 2 ,2x x
• You should always check all solutions with a calculator or by using the unit circle where appropriate.
• When solving equations you should also check that the solutions are defined for the domain of the tan, cot, sec and csc functions.
2. Find the values of x in degrees where o o2sec 1 1, 0 ,360x x
3. Solve: 4 cot + 3 = -2 cot - 8; 0,2
Second Degree Equations
• We solve second degree equations through techniques such as factoring(e.g., sin2 θ - 3sinθ + 2 = 0, for all θ)
or isolation and square root principles (e.g., tan2 θ - 3 = 0, for all θ).
4. Solve sin2 θ - 3sin θ + 2 = 0, for all θ (in degrees).
• Solution: This is similar to solving
x2 – 3x + 2 = 0
(x – 2)(x – 1) = 0
So, sin2 θ - 3sin θ + 2 = 0 factors to:
(sin θ – 2)(sin θ – 1) = 0
sin θ = 2 sin θ = 1
θ = θ =
• If the domain is real numbers, there are an infinite number of rotations on the unit circle in both a positive and negative direction.
• So to find all θ (in degrees) we write an expression for the values corresponding to θ = 90o
o o| 90 360 ,where kk
5. Solve: tan2 θ - 3 = 0, for all θ (in radians).
Solution: This can be solved by factoring a difference of squares
or by isolation and square root principles
isolation
square root
• You should realize that using only the principal square root in this equation causes a loss of roots.
2
2tan 3 0
2tan 3 0
2tan 3
tan 3
3
• Another common error occurs when you do not find all solutions for the given domain.
• Remember to focus on the given domain.
• In the equation above, the reference angle is __
and since there are two cases to consider (tangent being negative and positive), there are solutions in all four quadrants.
tan 3
6. Solve: where 0 ≤ θ < 2π22cos 5cos 2 0x x
7. Solve: sin2x + 5 sin x - 3 = 0; x ∈ (−π, 2π). Give exact solutions, or round to the nearest one hundredth.
8. Solve: 5 sec2x = 1- sec x; for all x in radians. Give exact solutions, or round to the nearest one hundredth.
#9 From text page 212
Page 211-213#3,4,5,7,16
