Fall 2016-MTH 06- D05-Trigonometry-Handout-5
1 Circles and Radian Measure
So far we have been using degrees as our unit of measurement for angles. However, there is
another way of measuring angles that is often more convenient. The idea is simple: associate
a central angle of a circle with the arc that it intercepts.
1.1 Radian measure
Consider a circle of radius r > 0. The circumference C of the circle is C = 2 π r, where π is an
irrational number almost equal to 3.14159265....
OA
B
AB = 14
C = π2
r
90◦
(a) θ = 90◦
OAB
AB = 12
C =π r
180◦
(b) θ = 180◦
O
A
B
AB = C = 2π r
360◦
(c) θ = 360◦
Figure 1.1 Angle θ and intercepted arc AB on circle of circumference C = 2πr
We see that a central angle of 90◦ cuts off an arc of length π2
r, a central angle of 180◦ cuts
off an arc of length π r, and a central angle of 360◦ cuts off an arc of length 2π r, which is the
same as the circumference of the circle.
360◦ = 2π radians
Formally, a radian is defined as the central angle in a circle of radius r which intercepts an
arc of length r. This definition does not depend on the choice of r.
O r
r
θ
θ = 1 radian = 180π
degrees ≈ 57.3◦.
The above relation gives us any easy way to convert between degrees and radians:
Degrees to radians: x degrees =(
π
180· x
)radians
Radians to degrees: x radians =
(180
π· x
)degrees
The statement θ = 2π radians is usually abbreviated as θ = 2π rad, or just θ = 2π when it is
clear that we are using radians. When an angle is given as some multiple of π, you can assume
that the units being used are radians.
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KerryTypewritten TextMTH 06 Trig Notes (for Ojakian's class)
Example
1. Convert 18◦ to radians.
2. Convert π9
radians to degrees.
Conversion between degrees and radians for some common angles:
Table 1 Commonly used angles in radians
Degrees Radians Degrees Radians Degrees Radians Degrees Radians
0◦ 0 90◦π
2180◦ 270◦
π
6120◦
7π
6300◦
45◦3π
4225◦ 315◦
π
3150◦
4π
3
11π
6
The default mode in most scientific calculators is to use degrees for entering angles. On
many calculators there is a button labeled✄
✂
�
✁DEG for switching between degree mode and ra-
dian mode. On some graphing calculators, such as the the TI-83, there is a✄
✂
�
✁MODE button for
changing between degrees and radians. Make sure that your calculator is in the correct angle
mode before entering angles. For example,
sin 4◦ = 0.0698 ,
sin (4 rad) = −0.7568 ,
so the values are not only off in magnitude, but do not even have the same sign. Using your
calculator’s✄
✂
�
✁sin−1 ,
✄
✂
�
✁cos−1 , and
✄
✂
�
✁tan−1 buttons in radian mode will of course give you the angle as
a decimal, not an expression in terms of π.
1.2 Arc length
We have seen that one revolution has a radian measure of 2π rad. Note that 2π is the ratio of
the circumference (i.e. total arc length) C of a circle to its radius r:
Radian measure of 1 revolution = 2π =2π r
r=
C
r=
total arc length
radius
Clearly, that ratio is independent of r. In general, the radian measure of an angle is the ratio
of the arc length cut off by the corresponding central angle in a circle to the radius of the circle,
independent of the radius.
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To see this, recall our formal definition of a radian: the central angle in a circle of radius r
which intercepts an arc of length r. So suppose that we have a circle of radius r and we place
a central angle with radian measure 1 on top of another central angle with radian measure 1,
as in Figure 1.2(a). Clearly, the combined central angle of the two angles has radian measure
1+1= 2, and the combined arc length is r+ r =2r.
r
r
11
2
r
(a) 2 radians
r/2
r/2
1/2
1
r
(b)12
radian
Figure 1.2 Radian measure and arc length
Now suppose that we cut the angle with radian measure 1 in half, as in Figure 1.2(b).
Clearly, this cuts the arc length r in half as well. Thus, we see that
Angle = 1 radian ⇒ arc length = r ,
Angle = 2 radians ⇒ arc length =
Angle = 12
radian ⇒ arc length =
and in general, for any θ ≥ 0,
Angle = θ radians ⇒ arc length = θ r ,
so that
θ =arc length
radius.
O r
s= rθ
θ
(a) Angle θ, radius r
O r′
s= r′θ
θ
(b) Angle θ, radius r′
Figure 1.3 Circles with the same central angle, different radii
We thus get a simple formula for the length of an arc:
In a circle of radius r, let s be the length of an arc intercepted by a central angle with
radian measure θ ≥ 0. Then the arc length s is:
s = rθ
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Example
1. In a circle of radius r = 2 cm, what is the length s of the arc intercepted by a central angle
of measure θ= 1.2 rad ?
2. In a circle of radius r =10 ft, what is the length s of the arc intercepted by a central angle of
measure θ= 41◦ ?
3. A central angle in a circle of radius 5 m cuts off an arc of length 2 m. What is the measure
of the angle in radians? What is the measure in degrees?
4. A rope is fastened to a wall in two places 8 ft apart at the same height. A cylindrical
container with a radius of 2 ft is pushed away from the wall as far as it can go while being
held in by the rope, as in the following figure which shows the top view. If the center of the
container is 3 feet away from the point on the wall midway between the ends of the rope, what
is the length L of the rope?
2
A
B
D CE3
4
4
θ
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Fall 2016-MTH 06- D05-Trigonometry-Handout-1
1 Trigonometric Ratios
Trigonometry is the study of the relations between the sides and angles of triangles. The
word “trigonometry” is derived from the Greek words trigono, meaning “triangle”, and metro,
meaning “measure”.
1.1 Angles
From elementary geometry:
An angle is formed by two rays (called the sides or arms of the angle), sharing a common
endpoint (called the vertex of the angle).
A degree, usually denoted by ◦ (the degree symbol), is a measurement of plane angle, repre-
senting 1360
of a full rotation.
Cvertex Aside
B
side
(a) angle ∠A,∠CAB
0◦
15◦
30◦
45◦60◦
75◦90◦
105◦120◦
135◦
150◦
165◦
180◦
195◦
210◦
225◦
240◦255◦270◦285
◦300◦
315◦
330◦
345◦
O
(b)angles in degree measurement
Instead of using the angle notation ∠A to denote an angle, we will sometimes use just a cap-
ital letter by itself (e.g. A, B, C) or a lowercase variable name (e.g. a, b, c). It is also common
to use letters (either uppercase or lowercase) from the Greek alphabet to represent angles. For
example:
Letters Name Letters Name Letters Name
A α alpha B β beta Γ γ gamma
Θ θ theta Φ φ phi Σ σ sigma
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Types of angles:
(a) An angle is acute if it is between 0◦ and 90◦.
(b) An angle is a right angle if it equals 90◦.
(c) An angle is obtuse if it is between 90◦ and 180◦.
(d) An angle is straight if it equals 180◦.
(a) angle (b) angle (c) angle (d) angle
Example:
• The angle 30◦ is
• The angle 45◦ is
• The angle 120◦ is
Types of pair of angles:
(a) Two acute angles are complementary if their sum equals 90◦.
(b) Two angles between 0◦ and 180◦ are supplementary if their sum equals 180◦.
(c) Two angles between 0◦ and 360◦ are conjugate (or explementary) if their sum equals
360◦.
∠A
∠B
(a) complementary
∠A
∠B
(b) supplementary
∠A∠B
(c) conjugate
Figure 1.1 Types of pairs of angles
Example:
• The angle 30◦ and the angle are complementary
• The angle is complementary to itself
• The angle 45◦ and the angle are supplementary
• The angle 300◦ is the conjugate angle of
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1.2 Triangles
A triangle has three sides and three interior angles. We have learned that the sum of the
interior angles in a triangle equals 180◦.
• Acute trianlges – triangles in which the measures of all three angles are less than 90◦.
• Right triangles – triangles in which the measure of one angle equals 90◦. Note that the
other two angles are acute angles whose sum is 90◦ (i.e. the other two angles are comple-
mentary angles).
• Obtuse triangles – triangles in which the measure of one angle is greater than 90◦.
Example:
For each triangle below, determine the unknown angle(s):
α
65◦ 70◦
53◦
α α α
3α
Triangles are similar if their corresponding angles are equal, and that similarity implies
that corresponding sides are proportional. We can also classify triangles according to the sides:
• Equilateral trianlges – triangles in which all three sides have the same length.
• Isosceles triangles – triangles in which two of the sides have the same length.
• Scalene triangles – triangles without two equal sides.
Example:
1. Draw an Equilateral Triangle. What are the three interior angles?
2. Draw a Right Isosceles Triangle (has a right angle 90◦ and also two equal angles). What
are the equal angles?
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1.3 Trigonometric Ratios for Acute Angles
In a right triangle, the side opposite the right angle is called the hypotenuse, and the other
two sides are called its legs.
A C
B
badjacent
a
op
posite
Pythagorean Theorem says:
a2 + b2 =c
hypo
tenu
se
Consider a right triangle △ABC, with the right angle at C and with lengths a, b, and c, as
in the figure on the right. For the acute angle A, call the leg BC its opposite side, and call
the leg AC its adjacent side.
The ratios of sides of a right triangle occur often enough in practical applications to warrant
their own names, so we define three basic trigonometric ratios of A as follows:
Name of ratio Abbreviation Definition
sine A sin A =opposite side
hypotenuse=
a
c
cosine A cos A =adjacent side
hypotenuse=
tangent A tan A =opposite side
adjacent side=
Notice that we have
tan A =sin A
cos A
The three basic trig ratios can be easily remembered using the acronynm SOHCAHTOA.
The other three trigonometric ratios, cosecant, secant and cotangent, are defined in terms of
the first three:
Name of ratio Abbreviation Definition
cosecant A csc A =hypotenuse
opposite side=
1
sin A
secant A sec A =hypotenuse
adjacent side=
1
cotangent A cot A =adjacent side
opposite side=
1
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Example
For the right triangle △ABC, find the values of all six trigonometric ratios of the acute angles
A and B.
A C
B
4
3?
Example Find the value of each trigonometric ratio.
1.
A C
B
8
6
sin A =
cos A =
tan A =
sinB =
cosB =
tanB =
2.
β
12α
13
sinα=
cosα=
tanα=
sinβ=
cosβ=
tanβ=
3.
β
4
α
2
sinα=
cosα=
tanα=
sinβ=
cosβ=
tanβ=
Question: Can the sine of an angle ever equal 2? How about cosine? tangent?
Note that when calculating the trigonometric functions of an acute angle A, you may use
any right triangle which has A as one of the angles.
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Example
1. Find the values of all six trigonometric ratios of 45◦.
A
B
C
1
1
1
1
45◦
2. Find the values of all six trigonometric ratios of 60◦.
A
B
C1 1
60◦ 60◦
30◦
2
3. Find the values of all six trigonometric ratios of 30◦.
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You may have noticed the connections between the sine and cosine, secant and cosecant,
and tangent and cotangent of the complementary angles in the examples. Actually we have
the following theorem.
Cofunction Theorem: If A and B are the complementary acute angles in a right triangle
△ABC, then the following relations hold:
sin A = cos B sec A = csc B tan A = cot B
sin B = cos A sec B = csc A tan B = cot A
Example
1. sin45◦ =
2. sin60◦ =
3. tan30◦ =
3. cos15◦ =
Given one trig ratio of an acute angle we can work out the others. In general it helps to draw
a right triangle to solve problems of this type.
Example
1. Suppose α is an acute angle such that sinα= 23. Find the values of the other trigonometric
ratios of α.
α
2
b
3
2. Suppose β is an acute angle such that cosβ= 15. Find the values of the other trigonometric
ratios of β.
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3. Suppose γ is an acute angle such that cotγ= 2. Find the values of the other trigonometric
ratios of γ.
KerryTypewritten TextSee workbook problems: p. 89, 90
KerryTypewritten Textp. 4,5: Problem 14 (Distance Formula)
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Fall 2016-MTH 06- D05-Trigonometry-Handout-2
1 Applying Right Triangles
We will use trigonometry to find the missing elements of a right triangle and solve application
problems involving right triangles.
1.1 Find missing elements of a right triangle
Basic strategy:
• Decide on the acute angle/ two sides with which we will work
• Identify the two sides as opposite, adjacent, or hypotenuse with respect to the angle we
chose
• Determine the trig ratio which relates the chosen angle and sides
• Write an equation using the ratio and solve the equation
Example
1. Find a and c in the given triangle.
2
c
a
60◦
30◦
2. Find a and b in the given triangle.
a
b
10
37◦
KerryTypewritten TextWorkbook problems: p.
KerryTypewritten Text92,93: #1, #2
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1.2 Solve application problems involving right triangles
Angle of elevation or depression:
object
object
angle of elevation
angle of depression
Example
1. A 30-ft ladder leans against a building so the the angle between the ground and the lad-
der is 62◦. How high does the ladder reach on the building?
h30
62◦
2. A blimp 4280 ft above the ground measures an angle of depression of 24◦ from its hori-
zontal line of sight to the base of a house on the ground. Assuming the ground is flat, how far
away along the ground is the house from the blimp?
24◦
4280
θ
x
KerryTypewritten TextWorkbook problems: p.93,94- problems #3,#4,#5
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3. A person standing 400 ft from the base of a mountain measures the angle of elevation
from the ground to the top of the mountain to be 25◦. The person then walks 500 ft straight
back and measures the angle of elevation to now be 20◦. How tall is the mountain?
h
500 400 x
20◦ 25◦
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Fall 2016-MTH 06- D05-Trigonometry-Handout-3
1 Trigonometric Functions and Cartesian Coordinates
1.1 Definition of a general angle
To define the trigonometric functions of any angle - including angles less than 0◦ or greater
than 360◦ - we need a more general definition of an angle.
We say that an angle is formed by rotating a ray−−→OA about the endpoint O (called the
vertex), so that the ray is in a new position, denoted by the ray−−→OB. The ray
−−→OA is called the
initial side of the angle, and−−→OB is the terminal side of the angle.
AOinitial side
B
term
inal
side
(a) angle ∠AOB
counter-clockwise
direction (+)
clockwise
direction (−) AO
B
(b) positive and negative angles
If the rotation is counter-clockwise then we say that the angle is positive, and the angle is
negative if the rotation is clockwise.
One full counter-clockwise rotation of−−→OA back onto itself (called a revolution), so that the
terminal side coincides with the initial side, is an angle of 360◦; in the clockwise direction
this would be −360◦. Not rotating−−→OA constitutes an angle of 0◦. More than one full rotation
creates an angle greater than 360◦. For example, notice that 30◦ and 390◦ have the same
terminal side, since 30+360= 390.
30◦
390◦
1.2 Trigonometric functions and Cartesian coordinates
Recall that the xy-coordinate plane consists of points denoted by pairs (x, y) of real numbers.
The first number, x, is the point’s x coordinate, and the second number, y, is its y coordi-
nate. The x and y coordinates are measured by their positions along the x-axis and y-axis,
respectively, which determine the point’s position in the plane. This divides the xy-coordinate
plane into four quadrants (denoted by QI, QII, QIII, QIV), based on the signs of x and y.
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x
y
0
QI
x > 0y> 0
QII
x < 0y> 0
QIII
x < 0y< 0
QIV
x > 0y< 0
(a) Quadrants I-IV
x
y
0
(2,3)
(−3,2)
(−2,−2)(3,−3)
(b) Points in the plane
x
y
0
θ
r
(x, y)
(c) Angle θ in the plane
Let θ be any angle. We say that θ is in standard position if its initial side is the positive
x-axis and its vertex is the origin (0,0).
Pick any point (x, y) on the terminal side of θ a distance r > 0 from the origin.
r =
We will define the trigonometric functions of θ as follows:
sin θ =y
rcos θ = tan θ =
csc θ = sec θ = cot θ =
As in the acute case, by the use of similar triangles these definitions are well-defined (i.e. they
do not depend on which point (x, y) we choose on the terminal side of θ). Also, notice that
|sin θ| ≤ 1 and |cos θ| ≤ 1, since |y| ≤ r and |x| ≤ r in the above definitions.
Notice that in the case of an acute angle these definitions are equivalent to our earlier defini-
tions in terms of right triangles: draw a right triangle with angle θ such that x = adjacent side,y = opposite side, and r = hypotenuse. For example, this would give us sin θ = y
r= opposite
hypotenuse
and cos θ = xr= adjacent
hypotenuse, just as before.
The following figure summarizes the signs (positive or negative) for the trigonometric func-
tions based on the angle’s quadrant:
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x
y
0
θr
hyp
oten
use
(x, y)
x
adjacent side
y
opposite side
(a) Acute angle θ
x
y
0
QI
0◦ < θ < 90◦QII
90◦ < θ < 180◦
QIII
180◦ < θ < 270◦QIV
270◦ < θ < 360◦
0◦
90◦
180◦
270◦
(b) Angles by quadrant
Figure 1.1
x
y
0
QI
sin +cos +tan +csc +sec +cot +
QII
sin +cos −tan −csc +sec −cot −
QIII
sin −cos −tan +csc −sec −cot +
QIV
sin −cos +tan −csc −sec +cot −
Figure 1.2 Signs of the trigonometric functions by quadrant
In general, if two angles differ by an integer multiple of 360◦ then they have the same initial
and terminal sides. Hence each trigonometric function will have equal values at both angles.
Angles such as these, which have the same initial and terminal sides, are called coterminal.
Example
1. Find five angles coterminal with 30◦.
2. Identify the angle that is not coterminal with the others
−120◦,600◦,−480◦,180◦,240◦
3. sin(390◦)= , cos(−315◦)= , tan(405◦)=
KerryTypewritten TextWorkbook problems: p.99,100: #5, #7
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The values of trigonometric functions of an angle θ larger than 90◦ can be found by using a
certain acute angle as part of a right triangle. That acute angle has a special name: if θ is a
nonacute angle then we say that the reference angle for θ is the acute angle formed by the
terminal side of θ and either the positive or negative x-axis.
Example
1. Let θ= 928◦.
(a) Which angle between 0◦ and 360◦ has the same terminal side
(and hence the same trigonometric function values) as θ ?
(b) What is the reference angle for θ ?
28◦
x
y
0
208◦
928◦
2. Draw the given angle in standard position and find the reference angle.
(a) 120◦ (b) 135◦ (c) −45◦
(a) −120◦ (b) 210◦ (c) 330◦
3. Find the exact values of all six trigonometric functions of 120◦.
x
y
0
120◦p
3
1
2
(−1,p
3)
60◦
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4. Find the exact values of all six trigonometric functions of 225◦.
x
y
0
225◦
1
1
p2
(−1,−1)
45◦
5. Find the exact values of all six trigonometric functions of 330◦.
x
y
0
330◦ 1
p3
2
(p
3,−1)
30◦
6. Find the exact values of all six trigonometric functions of 0◦, 90◦, 180◦, and 270◦.
x
y
0
0◦
(1,0)
90◦(0,1)
180◦
(−1,0)
270◦(0,−1)
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The following table summarizes the values of the trigonometric functions of angles between
0◦ and 360◦ which are integer multiples of 30◦ or 45◦:
Table 1 Table of trigonometric function values
Angle sin cos tan csc sec cot
0◦ 0 1 0 undefined 1 undefined
30◦ 12
p3
2
p3
32 2
p3
3
p3
45◦
60◦
90◦
120◦
135◦
150◦
180◦
210◦
225◦
240◦
270◦
300◦
315◦
330◦
Example
1. Suppose that cos θ=−45. Find the exact values of sin θ and tan θ.
x
y
0
θ3
4
5
(−4,3)
(a) θ in QII
x
y
0
θ
3
4
5
(−4,−3)
(b) θ in QIII
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2. Suppose that θ in the second quadrant and sin θ = 13
. Find the exact values of cos θ and
tan θ.
3. Suppose that θ in the fourth quadrant and tan θ = −2 . Find the exact values of sin θ andcos θ.
4. Suppose that θ in the third quadrant and csc θ =−1312
. Find the exact values of sin θ, cos θ
and tan θ.
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Fall 2016-MTH 06- D05-Trigonometry-Handout-6
1 Unit Circle and Trignometric Functions
1.1 Unit circle
The unit circle is the circle of radius 1 in the xy-plane consisting of all points (x, y) which satisfy
the equation x2 + y2 = 1.
x
y
s= rθ = θ1
θ
10
(x, y)= (cos θ,sin θ)x2 + y2 = 1
Each real number θ also corresponds to a central angle (in standard position) whose radian
measure is θ.
1.2 Unit circle and Trignometric Functions
Definitions of Trignometric Functions
Let θ be a real number and let (x, y) be the point on the unit circle corresponding to θ.
sin θ = y cos θ = x tan θ =y
x, x 6= 0
csc θ =1
y, y 6= 0 sec θ =
1
x, x 6= 0 cot θ =
x
y, y 6= 0
The unit circle has been divided into 8 equal arcs, corresponding to θ-values of 0, π4
, π2
, 3π4
,π, 5π4
, 3π2
, 7π4
and 2π.
x
y
(1,0)0
(p
22
,p
22
)(0,1)
(−p
22
,p
22
)
(−1,0)
(−p
22
,−p
22
)(0,−1)
(p
22
,−p
22
)
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The unit circle has been divided into 12 equal arcs, corresponding to θ-values of
0, π6
, π3
, π2
, 2π3
, 5π6
,π, 7π6
, 4π3
, 3π2
, 5π3
, 11π6
and 2π.
x
y
0◦
30◦
60◦90◦
120◦
150◦
180◦
210◦
240◦
270◦300◦
330◦
360◦
π
6
π
4
π
3
π
22π3
3π4
5π6
π
7π6
5π4
4π3
3π2
5π3
7π4
11π6
2π
(p3
2, 1
2
)
(p2
2,p
22
)
(
12,p
32
)
(
−p
32
, 12
)
(
−p
22
,p
22
)
(
−12,p
32
)
(
−p
32
,−12
)
(
−p
22
,−p
22
)
(
−12,−
p3
2
)
(p3
2,−1
2
)
(p2
2,−
p2
2
)
(
12,−
p3
2
)
(−1,0) (1,0)
(0,−1)
(0,1)
Example
1. Evaluate sinθ, cosθ, tanθ when θ = 0,π, π3
, π2
, 5π4
, 11π6
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2. Evaluate the following:
sin π6+cos 2π
3=
tan π4+cot 7π
4=
sin2 π3+cos2 π
3=
sin2 π4+cos2 π
4=
1+ tan2 5π6=
sec2 5π6=
tan 9π4+cos π
3=
sin 21π6
+cos(−2π3
)=
2sin π6
cos π6=
1−cos π3
2=
KerryTypewritten TextEvaluate trig functions at angles: 8pi/3, 8pi, 3pi, -17pi/4
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KerryPen
Fall 2016-MTH 06- D05-Trigonometry-Handout-7
1 Graphing the Trignometric Functions
The trigonometric functions can be graphed just like any other function, as we will now show.
In the graphs we will always use radians for the angle measure.
1.1 Unit circle
The unit circle is the circle of radius 1 in the xy-plane consisting of all points (x, y) which satisfy
the equation x2 + y2 = 1.
x
y
s= rθ = θ1
θ
10
(x, y)= (cos θ,sin θ)x2 + y2 = 1
We see from the unit circle that any point on the unit circle has coordinates (x, y)= (cos θ, sin θ),
where θ is the angle that the line segment from the origin to (x, y) makes with the positive
x-axis (by definition of sine and cosine). So as the point (x, y) goes around the circle, its y-
coordinate is sin θ.
We thus get a correspondence between the y-coordinates of points on the unit circle and
the values f (θ) = sin θ, as shown by the horizontal lines from the unit circle to the graph of
f (θ)= sin θ for the angles θ = 0, π6
, π3
, π2
:
θ
f (θ)
0
1
π
6π
3π
22π3
5π6
π
f (θ)= sin θ
π
6
π
3
π
2
01
1
x2+ y2 = 1
θ
Figure 1.1 Graph of sine function based on y-coordinate of points on unit circle
We can extend the above picture to include angles from 0 to 2π radians. This illustrates the
unit circle definition of the sine function.
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θ
f (θ)
0
−1
1
π
6π
3π
22π3
5π6
π 5π4
3π2
7π4
2π
f (θ)= sin θ
x
y
1
x2+ y2 = 1
θ
Figure 1.2 Unit circle definition of the sine function
Since the trigonometric functions repeat every 2π radians (360◦), we get, for example, the
following graph of the function y= sin x for x in the interval [−2π,2π]:
x
y
0
−1
1
π
4π
23π4
π 5π4
3π2
7π4
2π−π4
−π
2−3π4
−π−
5π4
−3π2
−7π4
−2π
y= sin x
Figure 1.3 Graph of y= sin x
To graph the cosine function, we could again use the unit circle idea (using the x-coordinate
of a point that moves around the circle), but there is an easier way. Recall from Section 1.5
that cos x = sin (x+90◦) for all x. So cos 0◦ has the same value as sin 90◦, cos 90◦ has the same
value as sin 180◦, cos 180◦ has the same value as sin 270◦, and so on. In other words, the
graph of the cosine function is just the graph of the sine function shifted to the left by 90◦ =π/2
radians, as in Figure 1.4:
x
y
0
−1
1
π
4π
23π4
π 5π4
3π2
7π4
2π−π4
−π
2−3π4
−π−
5π4
−3π2
−7π4
−2π
y= cos x
Figure 1.4 Graph of y= cos x
1.2 Amplitude and Period of Sine/Cosine Functions
Recall that the domain of a function f (x) is the set of all numbers x for which the function
is defined. For example, the domain of f (x) = sin x is the set of all real numbers, whereas the
domain of f (x) = tan x is the set of all real numbers except x =± π2
, ± 3π2
, ± 5π2
, .... The range
of a function f (x) is the set of all values that f (x) can take over its domain. For example, the
range of f (x) = sin x is the set of all real numbers between −1 and 1 (i.e. the interval [−1,1]),
the range of f (x) = cos x is the set of all real numbers between −1 and 1 (i.e. the interval
[−1,1]), whereas the range of f (x)= tan x is the set of all real numbers.
We know that −1≤ sin x ≤ 1 and −1≤ cos x≤ 1 for all x. Thus, for a constant A 6= 0,
−|A | ≤ A sin x ≤ |A | and −|A | ≤ A cos x ≤ |A |
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for all x. In this case, we call |A | the amplitude of the functions y= A sin x and y= A cos x. In
general, the amplitude of a periodic curve f (x) is half the difference of the largest and smallest
values that f (x) can take:
Amplitude of f (x) =(maximum of f (x)) − (minimum of f (x))
2
In other words, the amplitude is the distance from either the top or bottom of the curve to the
horizontal line that divides the curve in half.
x
y
0
|A |
−|A |
π
4π
23π4
π 5π4
3π2
7π4
2π
2 |A |
|A |
|A |
Figure 1.5 Amplitude = max−min2
=|A |−(−|A |)
2= |A |
A function f (x) is periodic if there exists a number p > 0 such that x+ p is in the domain of
f (x) whenever x is, and if the following relation holds:
f (x+ p) = f (x) for all x (1)
There could be many numbers p that satisfy the above requirements. If there is a smallest
such number p, then we call that number the period of the function f (x).
Example
Find the amplitude/period of each function and sketch the graph for x in [−2π,2π]
1. f (x)= 2sin x, x ∈ [−2π,2π]
Amplitude: 2; Period: 2π.
x
y
0
−2
2
π
4π
23π4
π 5π4
3π2
7π4
2π−π4
−π2−
3π4
−π− 5π
4− 3π
2− 7π
4−2π
y= 2sin x
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2. f (x)=−sin x
x
y
0
−1
1
π
4π
23π4
π 5π4
3π2
7π4
2π−π4
−π
2−3π4
−π−
5π4
−3π2
−7π4
−2π
3. f (x)= 2cos x
x
y
0
−2
2
π
4π
23π4
π 5π4
3π2
7π4
2π−π4
−π
2−3π4
−π−
5π4
−3π2
−7π4
−2π
4. f (x)=−3cos x
x
y
0
−3
3
π
4π
23π4
π 5π4
3π2
7π4
2π−π4
−π
2−3π4
−π−
5π4
−3π2
−7π4
−2π
KerryTypewritten TextWorkbook- p. 111: Problems 1,2,3,4
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Fall 2016-MTH 06- D05-Trigonometry-Handout-8
1 Basic Trigonometric Identities
So far we know a few relations between the trigonometric functions. For example, we know
the reciprocal relations:
1. csc θ =1
sin θwhen sin θ 6= 0
2. sec θ =1
cos θwhen cos θ 6= 0
3. cot θ =1
tan θwhen tan θ is defined and not 0
4. sin θ =1
csc θwhen csc θ is defined and not 0
5. cos θ =1
sec θwhen sec θ is defined and not 0
6. tan θ =1
cot θwhen cot θ is defined and not 0
Notice that each of these equations is true for all angles θ for which both sides of the equation
are defined. Such equations are called identities, and in this section we will discuss several
trigonometric identities, i.e. identities involving the trigonometric functions. These identities
are often used to simplify complicated expressions or equations. For example, one of the most
useful trigonometric identities is the following identity we have seen earlier:
tan θ =sin θ
cos θwhen cos θ 6= 0
To prove this identity, pick a point (x, y) on the terminal side of θ a distance r > 0 from the
origin, and suppose that cos θ 6= 0. Then x 6= 0 (since cos θ= xr), so by definition
sin θ
cos θ=
y
rx
r
=y
x= tan θ .
Note how we proved the identity by expanding one of its sides ( sin θcos θ
) until we got an expression
that was equal to the other side (tan θ). This is probably the most common technique for
proving identities. Taking reciprocals in the above identity gives:
cot θ =cos θ
sin θwhen sin θ 6= 0
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One of the most important trigonometric identities is the Pythagorean identity. Let θ be
any angle with a point (x, y) on its terminal side a distance r > 0 from the origin. By the
Pythagorean Theorem, r2 = x2 + y2 (and hence r =√
x2 + y2). For example, if θ is in QIII, then
the legs of the right triangle formed by the reference angle have lengths |x| and |y| (we use
absolute values because x and y are negative in QIII). The same argument holds if θ is in the
other quadrants or on either axis. Thus,
r2 = |x|2 + |y|2 = x2 + y2 ,
so dividing both sides of the equation by r2 (which we can do since r > 0) gives
r2
r2=
x2 + y2
r2=
x2
r2+
y2
r2=
( x
r
)2+
( y
r
)2.
Since r2
r2= 1, x
r= cos θ, and
yr= sin θ, we can rewrite this as:
cos2 θ + sin2 θ = 1
Note that we use the notation sin2 θ to mean (sin θ)2, likewise for cosine and the other
trigonometric functions.
From the above identity we can derive more identities. For example:
sin2 θ = 1 − cos2 θ
cos2 θ = 1 − sin2 θ
from which we get (after taking square roots):
sin θ = ±√
1 − cos2 θ
cos θ = ±√
1 − sin2 θ
In formula cos2 θ + sin2 θ = 1, dividing both sides of the identity by cos2 θ gives
cos2 θ
cos2 θ+
sin2 θ
cos2 θ=
1
cos2 θ,
so since tan θ= sin θcos θ
and sec θ = 1cos θ
, we get:
1 + tan2 θ = sec2 θ
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Likewise, dividing both sides of the Pythagorean identity by sin2 θ gives
cos2 θ
sin2 θ+
sin2 θ
sin2 θ=
1
sin2 θ,
so since cot θ = cos θsin θ
and csc θ = 1sin θ
, we get:
cot2 θ + 1 = csc2 θ
Example
1. Prove that cos2 θ tan2 θ = sin2 θ.
2. Prove that 5sin2 θ + 4cos2 θ = sin2 θ + 4.
3. Prove that tan θ + cot θ = sec θ csc θ .
4. Prove that1 + cot2 θ
sec θ= csc θ cot θ .
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5. Prove thattan2 θ + 2
1 + tan2 θ= 1 + cos2 θ .
6. Prove that1 + sin θ
cos θ=
cos θ
1 − sin θ.
7. Prove that1 − tan θ
1 + tan θ=
cot θ − 1
cot θ + 1
8. Prove that cos4 θ−sin4 θ = 1−2sin2 θ
KerryTypewritten TextWorkbook- p.
KerryTypewritten Text112: 1,2,3,4
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