MATH 105 – PLANE TRIGONOMETRY
MATHEMATICS 105
Plane Trigonometry
Chapter I
THE TRIGONOMETRIC FUNCTIONS
INTRODUCTION
The word trigonometry literally means triangle measurement. It is concerned with the measurement of the
parts, sides, and angles of a triangle. It further deals with six ratios that are determined by angles and
directed line segments. The main purpose of the study of trigonometry is to solve problems involving
triangles as what are found in astronomy, navigation and surveying.
With the development of trigonometry, trigonometric functions are associated with the lengths of the arcs
on the unit circle. It is now defined as a branch of Mathematics which is concerned with the properties and
applications of circular or trigonometric functions.
Plane Trigonometry, which is the concentration of this course, is restricted to the study of triangles lying in
a plane.
DIRECTED LINE SEGMENTS
Two concepts that are associated with a line segment are distance and direction. The distance from
one point to another, or the length of the line segment between the two points, is the number of times that
an accepted unit can be laid off along the given segment.
When a line segment is measured with a definite sense or direction from one endpoint to the other, the segment
is said to be directed line segment.
The distance between two distinct points on a directed line segment is called a directed distance.
distance.
A fundamental property of directed line is that if P1, P2 and P3 are any distinct points on the line, then the
following relation holds:
P1P3 = P1P2 + P2P3
On line L , we can assign an origin, a unit length and a positive direction. Figure 1, illustrates a line where the
distance between 0 and 1 corresponds to the unit length, the arrow at the right indicates positive direction. This line
is a one-dimensional coordinate system or a number line, where a one-to-one correspondence can be established.
-3
-2 -1 0
1 2
3
The directed distance from P1(x1) to P2(x2) regardless of the relative positions of 0, P1 and P2 is given by
P1P2 = x2 - x1
The undirected distance between P1 and P2 is the absolute vale of the directed distance between them.
| P1P2 | = | x2 - x1 |
EXERCISES:
Find the directed distance from P1 to P2 given the following:
No.
P1
P2
No.
P1
P2
1
6
-4
6
3.580
8.098
2
k
3k
7
¾
5
3
2½
5¾
8
2.37
-5.3
4
7
-8
9
7½
9.3
5
-b
a
10
-1.0007 2.083
Prepared by Mrs. Koni Gutierrez Cruz
Assistant Professor I
1
| Bataan Peninsula State University
50 m
14 m
MATH 105 – PLANE TRIGONOMETRY
THE RECTANGULAR COORDINATE SYSTEM
The Cartesian Coordinate System
Y
5 (x > 0, y > 0)
(x < 0, y > 0)
4
P(x,y)
3
II
I
2
Points in a plane are located using coordinate axes which consists
two perpendicular lines X’OX (horizontal) and Y’OY (vertical) which
intersect at the point O (refer to the figure at the right). The line X’OX is
called the x-axis and the line Y’OY is called the y-axis. Together they are
known as the Cartesian coordinate axes. The point of intersection is
called the origin.
1
2 3 4 5
O
X' -5 -4 -3 -2 -1
The coordinate axes divide the plane into four distinct regions called
quadrants marked I, II, III, IV (labelled counter clockwise). In quadrant I,
both x and y coordinates are positive (x > 0, y > 0), in quadrant II, xcoordinate is negative, y-coordinate is positive (x < 0, y > 0); in
quadrant III, both are negative (x < 0, y < 0) and in quadrant IV, xcoordinate is positive, y-coordinate is negative, (x > 0, y < 0).
III
(x < 0, y < 0)
X
-2
IV
-3
-4
(x > 0, y < 0)
-5
Y'
EXAMPLES:
1. Plot the points whose coordinates are:
a. P1 (5, 4)
b. P2 (4, -1)
c. P3 (0, 3)
d. P4 (-2, -5)
2. Give the coordinates of the following points and the quadrants where they are located:
a. five units to the right of the y-axis and two units below the x-axis
b. two units to the left of the y-axis and one unit below the x-axis
THE DISTANCE FORMULA
P1(x1,y1)
x1
d
(y2 - y1)
P2(x2,y2)
(x2 - x1)
x2
y2
Let the coordinates of two points be denoted by P1(x1, y1) and
P2(x2, y2), by Pythagorean theorem,
Y
y1
The distance between two points P1 and P2 can be
expressed in terms of their coordinates by the Pythagorean
theorem:
X
d = ( x 2 − x1 ) 2 +( y 2 − y1 ) 2
EXAMPLES:
1. Find the distance between two points:
P2(-10, 2)
a. P1(5, 14)
b. P1(2, 5)
P2(-1, 3)
If the two points in the distance formula are the origin and the point whose radius vector r we want, we find
that 0 0 . Consequently, squaring each member of this equation, a relation between
the coordinates and the radius vector r of P(x,y) is .
EXAMPLES:
1. If the abscissa of a point is 12 and its radius
vector is 13, find the values of the ordinate.
Solution: x = 12, r = 13,
y=?
12 13
144 169
169 144
25
5
Prepared by Mrs. Koni Gutierrez Cruz
Assistant Professor I
2. If the radius vector of a point in the second
quadrant is 5 and the ordinate is 3, find the
abscissa.
Solution: x = ?, r = 5, y = 3
3 5
9 25
16
4
Since the point is in the second quadrant,
3
we will use x = - 4
| Bataan Peninsula State University
MATH 105 – PLANE TRIGONOMETRY
EXERCISES:
1. Plot each point whose coordinates are given and find the radius vector of each. Furthermore, find the
distance between each pair of points.
A(3, 2)
B(7, - 5)
C(-8, 4)
D(- 3, -6)
2. Find the value or values of the one of x, y and r that is missing in each of the following:
a. x = 3,
y=4
b. x = -24
r = 25
c. x = 5
y = √11
d. x = -3
r = 5, y < 0
r = √27 x < 0
e. y = √2
ANGLES
Angles are what trigonometry is all about. This is where it all started, way back when. Early astronomers needed
a measure to tell something meaningful about the sun and moon and stars and their relationship between man
standing on the earth or how they are positioned in relation to one another. Angles are the input values for
trigonometric functions.
An angle is formed where two rays (straight line with an endpoint that extends
infinitely in one direction) have a common endpoint. This endpoint is called the
vertex. The two rays are called the sides of an angle – initial
initial and terminal sides.
A plane angle is to be thought of as generated by a revolving (in a plane) a ray from
the initial position to a terminal position.
is the initial side, while ray ON ON
is the terminal side.
In Figure 1, ∠NOP or ∠θ, ray OP OP
Writing Angle Names Correctly. An angle can be identified in several different ways:
•
•
•
Use the letter at the vertex of the angle.
Use the three letters that label the points – one on one side, the vertex and the
last on the other ray. Points are labelled with capital letters.
Use the letter or number in the inside of the angle. Usually, the letters used are
Greek or lowercase.
EXAMPLES:
Give all the different names that can be used to identify the angle shown in Figure 2.
Measure of an angle. An angle, so generated, is called positive if the direction of rotation is counterclockwise
and negative if the direction of rotation is clockwise. The common unit of measure of an angle is degree
denoted by ( ° ).
Classification of Angles. Angles can be classified by their size.
Acute Angle – an angle measuring less than 90°.
Right Angle – an angle measuring exactly 90°; the two sides are perpendicular
Obtuse Angle – an angle measuring greater than 90° and less than 180°.
Straight Angle – angle measuring exactly 180°.
Two angles can also be classified according to the sum of their measures. If the sum of the measures of the angles
is 90°, then the angles are called complimentary angles. If the sum is 180°, then the angles are called
supplementary
supplementary angles.
angles
4
Prepared by Mrs. Koni Gutierrez Cruz
Assistant Professor I
| Bataan Peninsula State University
MATH 105 – PLANE TRIGONOMETRY
When two line cross one another, four angles are formed. These angles which are opposite one another are
called vertical angles.
angles If two angles are vertical, then their measures are equal.
EXAMPLES:
1. If one angle in a pair of supplementary angles measure 80°, what does the other angle measure?
Answer:
The other measures 180° - 80° = 100°
2. What is the measure of the (a) complement (b) supplement of the angle in Figure 1 if θ is 38°?
Answer:
(a) complement is 90° - 38° = 52°
(b) supplement is 180° - 38° = 142°
EXERCISES:
1. Find the measure of the complement of an
angle whose measure is:
a. 37°
b. 58°
c. x°
d. 90° - y°
e. 37° + R°
2. Determine the measure of the supplement of an
angle whose measure is:
a. 75°
b. 85°
c. x°
d. 90° + y°
e. 113.76° + R°
ANGLE IN STANDARD POSITION
OSITION
An angle is in standard position with reference to a system of rectangular coordinate axes if its vertex is at the origin
and its initial side lines along the positive ray of the X axis.
If an angle is in standard position, the angle is said to be in the quadrant in which the terminal side lies. Hence, an
acute angle is in the first quadrant; an obtuse angle is in the second quadrant; an angle of 215° is in the third
quadrant; an angle of 330° is in the fourth quadrant.
If the terminal side of an angle in standard position coincides with one of the coordinate axes, the angle is a
quadrantal angle.
angle An angle of 90° and any angle which is an integral multiple of 90° is a quadrantal angle. Two
angles are coterminal if they are in standard position and have the same terminal sides. Thus, of 150°, 510°, and 210° are coterminal angles.
EXERCISES:
1. Construct the following angles in standard position and determine those which are coterminal:
a. 125°
f. -370°
b. 210°
g. -955°
c. -150°
h. -870°
d. 385°
i. 595°
e. 930°
j. -210°
2. Give three other angles coterminal with
a. 125°.
b. 45°
c. 165°
d. 590°
e. -30°
f.
g.
h.
i.
j.
75°
-120°
-60°
270°
135°
5
Prepared by Mrs. Koni Gutierrez Cruz
Assistant Professor I
| Bataan Peninsula State University
MATH 105 – PLANE TRIGONOMETRY
DEFINITION OF THE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE IN STANDARD POSITION
The six ratios on which the subject of trigonometry is based are formed by using the abscissa, the ordinate, and the
radius vector of a point on the terminal side of an angle in standard position. Since the values of these ratios
depend on the values of the abscissa, the ordinate, and the radius vector, and these in turn depend on the size of
the angle, each ratio is a function of the angle.
We shall now give the definitions of these six functions and, beside each definition, the abbreviated form in which it
is usually written. These definitions and abbreviated forms should be memorized. If the angle θ is in standard
position, then
The sine of angle θ
!"#$% & '
= # !() *%+$ & '
sin θ =
,
cos θ =
.
tan θ =
,
.
The cosine of angle θ
#-)+!))# & '
= # !() *%+$ & '
The tangent of angle θ
!"#$% & '
= #-)+!))# & '
The cosecant of angle θ
=
# !() *%+$ & '
!"#$% & '
csc θ = ,
The secant of angle θ
=
# !() *%+$ & '
#-)+!))# & '
sec θ = .
The cotangent of angle θ
#-)+!))# & '
!"#$% & '
.
= cot θ = ,
It should be noted that the cosecant is the reciprocal of the sine, the secant the reciprocal of the cosine, and the
cotangent the reciprocal of the tangent.
The values of the functions of an angle are not affected by the position of the point P on the terminal sides as can
seen by the following consideration. If P and P’ are any two points on the terminal side of θ, and if QP and Q’P’ are
perpendicular to the X axis, then the triangles OQP and OQ’P’ are similar; hence, corresponding sides are
/'
/1'1
proportional. Thus, 0' = 0'1 , and we see that the sine ratio has the same value regardless of the point on the
terminal side that is used.
In a similar way it can be shown that the values of the other five functions are not affected by the position of P.
EXAMPLES:
1. If the point (5,-12) is on the terminal sides of an angle θ that is in standard position, find sin θ, cos θ, and tan θ.
Solution:
By use of the relation between the abscissa, ordinate, and radius vector of a point, we see that =
5 12 25 144 169. Hence, the radius vector is 13, and we have
,
sin θ = =
23
,
34
.
5
cos θ = = 34 ,
,
tan θ = . =
23
5
.
EXERCISES:
1. Determine the values of the trigonometric functions of angle θ if P is a point on the terminal side of θ and the
coordinates of P are:
a. P(3, 4)
b. P(-3, 4)
c. P(-1, -3)
d. P(√2, 1
e. P(2, -3)
6
Prepared by Mrs. Koni Gutierrez Cruz
Assistant Professor I
| Bataan Peninsula State University
MATH 105 – PLANE TRIGONOMETRY
ALGEBRAIC SIGNS OF THE TRIGONOMETRIC FUNCTIONS
The algebraic signs of the functions of an angle depend on the signs of the
abscissa and the ordinate, since the radius vector is always positive. For
example, since the abscissa of every point in the second quadrant is negative
and the ordinate is positive, any function which uses the abscissa is negative
while all others are positive. However, for any point in the third quadrant, the
abscissa and the ordinate are both negative; hence, the cotangent are positive
since each is the ratio of two negative numbers. The other functions are
negative.
Figure at the right shows functions are positive for each quadrant; all other
functions are negative.
FUNCTIONS OF 30°
30°, 45°
45° AND THEIR MULTIPLES
The computation of the values of trigonometric functions of angles in general is beyond the scope of an elementary
trigonometry book. We can, however, find the values of functions of 30°, 45°, 60°, and their multiples by use of
some theorems from geometry and the definition of the trigonometric functions.
We shall first consider a 45° angle in standard position as shown in Fig. 1-11. From plane geometry, we know that a
45° right triangles is isosceles. If each of the equal sides is 1 unit long, by the Pythagorean Theorem the hypotenuse
is √2 units. Hence, the coordinates and radius vector P are as shown in Fig. 1-11. The definition of the trigonometric
functions gives
Sin 45° =
cos 45° =
3
√
3
√
3
=
√
,
cot 45° = = 1,
=
√
,
Sec 45° =
3
tan 45° = = 1,
3
3
√
3
= √2 ,
csc 45° =
√
3
= √2 ,
In order to find the functions of 30° and 60°, we use the theorem from plane geometry which states that, in a 30°
right triangle, the side opposite the 30° angle is half as long as the hypotenuse. Hence, if we put the 30° angle in
standard position, the coordinates and the radius vector of P are as shown in Fig. 1-12, since the Pythagorean
theorem gives x 1 2 and x = √3. Now by use of the definition, we have
3
Sin 30° = ,
cos 30° =
tan 30° =
√4
3
√4
cot 30° =
=
=
√4
3
= √3,
√4
Sec 30° = =
4
√4
,
4
4
,
csc 30° = = 2 .
3
In other to find the values of the functions of a 60° in standard position as shown in Fig. 1-13. The other acute angle
of the 60° right triangles is 30°. hence, the values of the coordinates and radius vector of P are as shown. We need
now only apply the definitions to see that
Sin 60° =
cos 60° =
tan 60° =
√4
,
3
3
√4
=
√4
,
4
Sec 60° = 3 = 2 ,
√4
3
cot 60° =
= √3,
csc 60° =
√4
=
√4
4
.
The values of the trigonometric functions of integral multiples of 30°, 45° can be found by making use of the values
of the functions of 30°, 45°, and 60° and the reference angle. If an angle is in standard position, then the acute
angle between the terminal side and the X axis is called the reference angle. Thus, the reference angle of 150° is
7
30° and so is the reference of 210°. The reference angle of 225° is 45° and that 300° is 60°.
Prepared by Mrs. Koni Gutierrez Cruz
Assistant Professor I
| Bataan Peninsula State University
MATH 105 – PLANE TRIGONOMETRY
EXAMPLES
XAMPLES:
1. Find the values of the six trigonometric functions of 150°.
Solution
Draw an angle of 150° in standard position. The angle formed by the terminal
side of the 150° angle and the negative ray of the X axis is the supplement of 150°,
or 180° - 150° = 30°.
3
sin 150° = ,
csc 150° = =2 .
cos 150° =
3
2√4
sec 150° = 2√4 =
3
tan 150° = 2√4 =
2√4
4
, cot 150° =
2√4
3
2√4
4
,
= √3,
2. Evaluate 2 sin 225° - tan 135°
Solution:
2√ 2(;) + 1
2sin 225° - tan 135° =
2
=
=
- (-1)
2
FUNCTIONS OF QUADRANTAL
UADRANTAL ANGLES
A quadrantal angle was defined as an angle that is in standard position and has its terminal side along a coordinate
axis. It follows that the abscissa or the ordinate of a point on the terminal side is zero, and as result has to be used
in finding some of the values of the functions of the angle. It must be remembered that zero divided by any number
is zero, and that no number can be divided by any number is zero, and that no number can be divided by zero. It
follows from these two facts that two of the function of a quadrantal angle are zero and that there is no number to
represent another two of the functions. The functions of 90° can be obtained from Fig. 1-16 and are given below
3
<
cot 90° = 3 = 0,
sin 90° = 3 = 1,
<
3
cos 90° = 3 = 0,
sec 90° = < = undefined,
tan 90° = =undefined,
csc 90° = =1.
3
<
3
3
Functions of other quadrantal angles can be found in similar manner.
VALUES OF TRIGONOMETRIC FUNCTIONS FOR SOME FAMILIAR ANGLES
Angle α
Degree
Radian
0
0°
?
30°
6
?
45°
4
?
60°
3
?
90°
2
?
180°
3?
270°
2
2?
360°
sin α
cos α
tan α
csc α
sec α
cot α
0
1
0
∞
1
1
2
√2
2
√3
2
1
∞
√3
2
√2
2
1
2
√3
3
2
2√3
3
√3
1
√2
√2
1
0
∞
0
-1
0
∞
-1
∞
-1
0
∞
-1
∞
0
0
1
0
∞
1
∞
√3
2√3
3
1
2
∞
√3
3
0
8
Prepared by Mrs. Koni Gutierrez Cruz
Assistant Professor I
| Bataan Peninsula State University
MATH 105 – PLANE TRIGONOMETRY
EXERCISES
1. Find the functions of the following:
a. 30°, 135°, 240°
b. 45°, 120°, 330°
c. 60°, 315°, 210°
d. 300°, 225°, 150°
2. Evaluate the combination of functions of angles given in each of the following.
@AB 3C<° D@ABE<°
a. sin 90°cos 30° + cos 90° sin30°
c.
32@AB 3C<° @ABE<°
b. cos 180°cos 60° + sin 180° sin 60°
32@AB ;<° @AB;5°
d.
@AB 4<<°2 @AB435°
3. Verify the following statements:
a.
b.
c.
d.
e.
cos 45° - sin 135° = cos 90°
sin 60° + cos 240° = tan 225°
sin 30° + cos 150°+ tan 60° = sec 300°
sec 210° - tan 330°= sin 150° + cos 30°
sin 240° = 2sin 60°cos 240°
cos 150° 2cos 45° cos 330° - sin 135° sin30° tan 315°
cot 330° - csc 150° sin 120° cos 300°- 2
2 tan 150° tan 120° 1 tan 210°
f.
g.
h.
4. Identify each of the following as true or false.
a.
b.
c.
d.
sin 150° cos 30°tan 210°
sin 120° 2 sin 120° cos 300°
cos 300° cos 30° - sin 330°
RSB 3<°
RSB 4<°
e.
f.
g.
4
3
tan 150° tan 30°
5
cos 90° cos 45° sin 135°
tan 300° cot 120° tan 225°
RELATIONS BETWEEN DEGREES AND RADIANS
A degree (° ) is defined as the measure of the central angle subtended by an arc of a circle equal to
3
of the
circumference of the circle. The degree is divided into 60 minutes and a minute is divided into 60 seconds.
3
3
3
A minute ( ‘ ) is E< of a degree; a second ( “ ) is E< of a minute or 4E<< of degree.
4E<
EXAMPLES:
1. Express each angle measures in degrees, minutes and seconds.
a. 18.5° = 18° + 0.5°
b. 6°8
31
;
= 6° + 8.25’
E<1
U
3°
E<11
.25’T 31 U
= 18° + 0.5°T
= 18°30’
= 6° + 8’ +
= 6° 8’ 15”
E<1
= 75° + 0.12° + 0.005° = 75° + 0.125°T 3° U = 75° + 7.5’
c. 75.125°
E<11
U
31
= 75° + 7’+ 0.5’T
= 75° 7’ 30”
2. Express each angle measures in decimal degrees:
1°
a. 45° 12’ 36” = 45° + 12’T U+ 36”T
60′
1°
U = 45° + 0.2° + 0.01° = 45.21°
3600′
1°
b. 86° 15.6’ = 86° + 15.6’T U = 86° + 0.26° = 86.26°
60′
A radian (rad) is defined as the measure of the central angle subtended
by an arc of a circle equal to the radius of the circle.
The circumference of a circle = 2?(radius) and subtends an angle of
360°.
2? radians = 360°
? radian = 180°
1 WXYWZ 3C<°
[
57.296° 57°17′45"
[
1 degree = 3C< WXYWZ 0.017453 WX
Prepared by Mrs. Koni Gutierrez Cruz
Assistant Professor I
9
| Bataan Peninsula State University
MATH 105 – PLANE TRIGONOMETRY
EXAMPLES:
1. Convert the following into radian. Express answer as multiple of π.
a. 30° 30°T
[
3C<°
U
4<[
3C<
[
2. Express the following in terms of degrees:
a.
[
[ 3C<
T
;
;
[
U 45°
b.
_[
E
3. Convert the following into degree measure:
_
a.
? WX
(105°)
b.
c.
? WX
4
5
4E
? WX
(120°)
^[
;<
_[ 3C<
E
T
[
U 210°
EXERCISES:
1. Express the following in degrees, minutes and seconds:
a. 62.4° (62°24’)
d.
b. 23.9° (23°54’)
e.
c. 29.23° (29°13’48”)
2. Express the following in degrees (rounded to hundredths):
a. 78°17’
(78.28°)
d.
b. 58°22’16” (58.37°)
e.
c. 120°30’45” (120.51°)
3
[
b. 40°30’40.5°T3C<°U E
37.47°
(37°28’12”)
34.125°(34°7’30”)
5 right angles
8.5 revolutions
d. 0.2130 WX
353
e.
? WX
4E<
(450°)
(3060°)
(12°12’20”)
(75°30’)
(25°)
4. Convert the following into radian measure. Express answer as multiple of `:
a. 90°
d. 150°
b. 216°
e. 5°37’30”
c. 7°12’
5. Express each angle in terms of radians rounded to hundredths.
a. 8°
d. 44°44’44”
b. 24°31’
e. 71.71°
c. 38°19’20”
Assignment:
Ref. Book: Plane Trigonometry by Barcelon, et al.
p. 39 A to D.
10
Prepared by Mrs. Koni Gutierrez Cruz
Assistant Professor I
| Bataan Peninsula State University