Download Lesson I The Trigonometric Ratios of Angles 00 0

5enLor 3 Pre-Calcultis Mathematics
Module 2, Lesson I
Lesson I
The Trigonometric Ratios of Angles
00
0
3600
Objectives
1. Given an angle you will be able to draw it in standard
position.
2. Given an angle in standard position and a point on it
terminate side, you will be able to state the sine, cosine, and
tangent of the angle in terms of the coordinates of this point.
3. Using a calculator, you will be able to find the sine, cosine,
and tangent of any angle from 00 to 3600.
In your study of trigonometry, you have studied angles between
00 and 180°. We will review this and extend it to 360. We will
review angles in standard position.
Definition
An angle is in standard position if:
• its first side called the initial side is a ray from the origin
along the positive x-axis
• its second side called the terminal side is any ray from the
origin
• the angle is measured from the initial to the terminal side in a
counterclockwise direction
Quadrant I
Quadrant II
V
initial side
x
0 is in standard position. The
measure of 0 is between 00
and 900.
initial side
& is in standard position. The
measure of 0 is between 00
and 180°.
_
4
iodule 2, Lesson I
Quadrant III
Quadrant
Senior 3 ?re-Calculus Mathematics
f\T
J
I
I
initial side
X
::1side
8 is in standard position. The
measure of 8 is between 180°
and 270°.
8 is in standard position. The
measure of 0 is between 270°
and 360°.
j
We will define the trigonometric ratios in terms of the
coordinates of any point P(x, y) on the terminal side of an angle
in standard position.
Definitions
If 8 is in standard position and P(x, y) is any point on the
terminal side of 8, then by the Pythagorean Theorem
r=x2+y2
P(x,y)
is called the radius
vector and its length is
always positive.
r
J
I
j
I
I
]
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Senior 3 Pre-Calculus Mathematics
1oduIe 2, Lesson 1
5
If P(x, y) is any point on the terminal arm of an angle in
standard position.
y coordinate
rathus vector
x coordinate
cos6=
radius vector
y coordinate
tan
x coordinate
y
r
—
x
—
—
—
r
=
y
x
If P(x, y) is in Quadrant I, the measure of 0 < 900, x
and y are
positive, and all trigonometric ratios of angles are
positive.
P(x, y)
x
For any angle 0 where 9
sin
=
cos
=
90° and x> 0 and y> 0,
will be positive (>0)
x
tan 6? =
<
—
.
will be positive (>0)
will be positive (>0)
:oduIe 2, Lasson I
senior 3 Pre-Calculus Mathematics
If P(x, y) is in Quadrant II, the measure of 9 is between 900 and
180°, and the x-value of P(x, y) is negative. Any trigonometric
ratio in Quadrant II whose ratio is defined using x will be
negative.
P(x, y)
For any angle where 90° <8
sin 61 =
cos 6 =
tan 61 =
<
180° and x < 0 and y> 0,
will be positive (>0)
-
will be negative (<0)
will be negative (<0)
If P(x, y) is in Quadrant III, the measure of 9 is between 180°
and 270°, and the x- and y-value of P(x, y) are negative.
y
P(x, y)
1oduIe 2, Lesson 1
Senior 3 Pre-Calculus Mathematics
For any angle where 1800 <8
sin 6 =
-
7
<
270° and x < 0 and y
<
0,
will be negative (<0)
cos 6 =
will be positive (<0)
tan 6 =
wifi be positive (<0)
If P(x, y) is in Quadrant IV, the measure of 8 is between 270°
and 360°, and the x-value is positive and the y-value is negative.
y
—,,.
x
P(x, v)
For any angle where 270° <8
<
360° and x> 0 and y
sin 6=
will be negative (<0)
cos 6 =
wifi be positive (>0)
<
0,
tan6=wfflbe negative (<0)
Notice when you have 8 in any one of the four quadrants, each
trigonometric function will be positive in two quadrants and
negative in the other two.
B
1iodule 2, Lesson I
.5enior 3 PR-Calculus Mathematics
xampIe I
P(x, y) is a point on the terminal side of angle 0, in standard
position. Determine sin 8, cos 8, and tan 8.
Solution
a) P(3, 4)
P(34)
r=,./32+42
sin
/
4
=
y4
=
r 5
—
—
x3
cos6=—=—
r 5
y4
tan6=—=—
x3
3
b) P(—3, 4)
P(—3, 4)
r
=
J(3)2 42
=
4
5
—3
=
5
4
4
y
tan6=—=—=—-x —3
3
sin t9= y
r
x
cosi9=
r
—
4\
=
c)
p x
—3
P(—3, —4)
y
4—.
—4
-3
4N
/
P(—3, —4)
r=
\/(_3)2
—
—
—
-.-.——
.,/ =5
+ (_4)2
y
=
=
—4
5
—3
5
—4
—3
sin=—=-——
r
x
=
r
tan 9 = y
x
cost9=
—
—
—
—
=
4
3
—
5
Senior 3 Pre-Calculus Mathematics
2oduIe 2, Lesson 1
d) P(3, —4)
19
2
[
r=3+(—4) =-i2
o
N
4
y—
sin6?=_=
3
P(3, —4)
Example 2
Use your calculator to find each of the following and comme
nt
on the sign of the ratio.
a) cos 2000
b) tan 315°
c) sin 150°
d) tan 240°
e) cos 275°
f sin 2500
g) cos 150°
g) sin 315°
Solutions (answers rounded to 5 places)
a) cos 200° = —0.93969
x-coordinate is negative in Quadrant III
b) tan 315° = —1.0
y-coordinate is negative in Quadrant IV
c)
sin 150° = 0.5
y-coordinate is positive in Quadrant II
d) tan 240° = 1,732105
x- and y.coordinates are both negative in Quadrant III
so the
ratio will be positive
e) tan 275° —11.43005
y-coordinate is negative in Quadrant IV
L
I
sin 250° = —0.93969
y-coordmate is negative in Quadrant III
g) cos 150° —0.86603
x-coordinate is negative in Quadrant II
h) sin 315° = —0.70711
y-coordinate is negative in Quadrant IV
10
Module 2, Lesson I
Senior 3 Pre-Calculus Mathematics
Assignment I
1. P(x, y) is a point on the terminal side of angle 8 in standard
position. Determine sin 8, cos 8, and tan 8 for the following
points. Draw a sketch.
a) (—5, 12)
b) (—5, —12)
c) (5, —12)
d) (—4, —3)
e) (24, —7)
f) (—8, 15)
g) (8, —15)
h) (5, 3)
i) 8, —2)
j) (—3, —7)
2. Use your calculator to find the value of each of the following
functions and comment on the sign of the ratio. Round to five
decimal places.
a) cos 181°
b) sin 255°
c) tan 340.4°
d) sin 261°
e) cos 224°
1) tan 152.2°
g) cos 121.5°
h) sin 332°
i) tan 271.6°
]
3. Determine the quadrants in which P(x, y) may lie under the
following conditions.
j
a) sin 9> 0
b) cos 8> 0
d) sin 6
e) cos 9
<
0
<
c) tan 8> 0
0
0 cos&>Oandsjn9 <0
.1
a