math 31 lessons chapters 6 & 7: trigonometry 1. trigonometry basics
TRANSCRIPT
MATH 31 LESSONS
Chapters 6 & 7:
Trigonometry
1. Trigonometry Basics
Section 6.1: Functions of Related Values
Read Textbook pp. 250 - 267
A. Standard Position
Angles are created by rotating an “arm” from the
positive x-axis, which is called the initial arm.
Where the angle ends is called the terminal arm.
Counterclockwise angles are positive.
Clockwise angles are negative.
e.g.
Sketch the angle +200 in standard position.
y
x
y
xstart
Angles in standard position are measured from the positive x-axis, which is the initial arm.
y
xstart
It is useful to include the major angles at each axis. The clockwise direction is positive.
0, 360
90
180
270
y
xstart
Now, we sweep the arm 200 counterclockwise, since it is a positive angle.
0, 360
90
180
270
200
e.g.
Sketch the angle 310 in standard position.
y
x
y
xstart
Angles in standard position are measured from the positive x-axis, which is the initial arm.
It is useful to include the major angles at each axis. This time, we want the negative angles (clockwise).
y
xstart 0, 360
270
180
90
Now, we sweep the arm 310 clockwise, since it is a negative angle.
310
y
xstart 0, 360
270
180
90
Radians
Another measure of angle, apart from degrees, is radians.
To convert from radians to degrees (and vice versa),
use the following information:
2 radians = 360
or, radians = 180
y
x
These are the major angles at each axis in radians.
0, 2
2
2
3
Converting from Radians to Degrees
e.g. Convert radians to degrees.4
7
4
7 4
1807
300
Since radians = 180, you can simply substitute 180 wherever you see .
Converting from Radians to Degrees
e.g. Convert 252 to radians.
180
radians252
degrees goes on the bottom, since it must cancel
Recall, radians = 180
180
radians252
radians5
7
B. Trig Ratios
When we define the trigonometric ratios, we will use
a circle (rather than a triangle).
In this way, we can deal with angles that are
bigger than 180 (as well as negative angles).
Primary Trig Ratios
Consider a circle of radius r.y
x
r
We consider any point P
that is on the circumference
of the circle.
Its general coordinates
will be (x, y).
P (x, y)
We can create a triangle
with height y and base x.
The hypotenuse will be r,
since it represents the
radius of the circle.
r y
x
P (x, y)
Using the Pythagorean
theorem,
r y
x
P (x, y)
222 yxr
22 yxr r is the radius of the circle, so it must always be positive (r > 0)
We can now use
“Soh Cah Toa” to define
each primary trig ratio. r y
x
P (x, y)
“Soh Cah Toa”
r y
x
P (x, y)
""
""sin
hyp
opp
r
ysin
“Soh Cah Toa”
r y
x
P (x, y)
""
""cos
hyp
adj
r
xcos
“Soh Cah Toa”
r y
x
P (x, y)
""
""tan
adj
opp
x
ytan
Reciprocal Trig Ratios
sin
1csc
cos
1sec
tan
1cot
Ex. 1 Evaluate cos and csc if
Answer in exact values. Do not find the angle.
Try this example on your own first.Then, check out the solution.
2
,12
5tan
0
Determine the quadrant of the angle
2
y
x
2
x
The angle is in quadrant 2, where x is negative and y is positive
Sketch the triangle
Thus, x = -12 and y = 5
12
5tan
x
y Remember, x is negative and y is positive
12
5tan
r
x = -12
y = 5
Find r
r
x = -12
y = 5 222 yxr
22 yxr
22 512
13
Find cos
r = 13
x = -12
y = 5 r
xcos
13
12cos
Find csc
r = 13
x = -12
y = 5
sin
1csc
ry1
y
r
5
13
C. Reference and Coterminal Angles
Reference Angle
The reference angle is the acute angle (< 90)between the terminal arm and the nearest x-axis.
Reference angles are always positive.
e.g.
What is the reference angle for 260?
First, we sketch the angle.
y
xstart 0, 360
90
180
270
260
The reference angle is the angle between the terminal arm
and the nearest x-axis.
y
x18080
270
The reference angle is 80
Coterminal Angles
Two angles that have the same terminal arm
are called coterminal angles.
y
x
y
x
and are coterminal.
Note:
For any given reference angle (e.g. 50),there are an infinite number of coterminal angles.
y
x50
Note:
For any given reference angle (e.g. 50),there are an infinite number of coterminal angles.
y
x50
The smallest positive angle to the terminal arm (130)is called the principal angle.
y
x50
130
The next positive angle with the same terminal arm is constructed by adding 360 to the principal angle.
y
x50
130 + 360 = 490
The first negative angle with the same terminal arm is constructed by subtracting 360 from the principal angle.
y
x50
130 - 360 = -230
In general, we can find all coterminal angles by adding
or subtracting multiples of 360 from the principal angle.
i.e. If 1 and 2 are coterminal, then
2 = 1 + (360) n , where n I
n belongs to the integers.
Thus, n ... , -3, -2, -1, 0, 1, 2, 3, ...
If 1 and 2 are coterminal, then
2 = 1 + (360) n , where n I
or in radian form,
2 = 1 + 2 n , where n I
D. Exact Trig Values (Using Special Triangles)
It is crucial that you remember the
exact values for the trig ratios of the following angles:
30 45 60
To do so, we need to use special triangles.
60 "Soh Cah Toa"
60
32 2
1 1
3sin 60 = 2
cos 60 = 1 2
3tan 60 = 1
30 "Soh Cah Toa"
30
32 2
1 1
3cos 30 = 2
sin 30 = 1 2
3tan 30 = 1
45 "Soh Cah Toa"
45
2
1
1
2
sin 45 = 1
tan 45 = 1
2
cos 45 = 1
E. CAST
This is a simple but effective way to remember the signs
of all trig ratios in each quadrant.
C
AS
T
12
3 4
C
All +S
T
12
3 4
sin + cos + tan +
C
ASine +
T
12
3 4
sin + cos tan
C
AS
Tan +
12
3 4
sin cos tan +
Cos +
AS
T
12
3 4
sin cos + tan
F. Unit Circle
If we define the trig circle with a radius of 1 unit,
called the unit circle, then finding exact values for
the trig ratios is much more straightforward.
1
1
“Soh Cah Toa”
1 y = sin
x""
""sin
hyp
opp
yy
1
sin
The sine ratio is simply the y-coordinate.
“Soh Cah Toa”
1 y
x = cos ""
""cos
hyp
adj
xx
1
cos
The cosine ratio is simply the x-coordinate.
“Soh Cah Toa”
r y
x
P (x, y)
""
""tan
adj
opp
x
ytan
The tangent ratio remains y over x.
In general,
r y = sin
P (sin , cos )
x = cos
Building the unit circle ...
(1, 0)
(0, 1)
(-1, 0)
(0, -1)
Start with the axes.
(1, 0)
(0, 1)
(-1, 0)
(0, -1)
2
1,
2
330
6
2
2,
2
245
4
2
3,
2
160
3
Next, add the special triangle ratios in quadrant 1.
Remember, x = cos and y = sin
(1, 0)
(0, 1)
(-1, 0)
(0, -1)
2
1,
2
330
6
2
2,
2
245
4
2
3,
2
160
3
It is crucial that you memorize this special triangle.
Ex. 2 Evaluate the following exactly (without your calculator):
Try this example on your own first.Then, check out the solution.
2tancos5
2sin3
Convert to degrees (if you need to)
2tancos5
2sin3
1802tan180cos52
180sin3
Recall, radians = 180
2tancos5
2sin3
1802tan180cos52
180sin3
360tan180cos590sin3
Evaluate the special angles
using the unit circle
(1, 0)
(0, 1)
(-1, 0)
(0, 1)
2
1,
2
330
6
2
2,
2
245
4
2
3,
2
160
3
190sin
Recall, sine is the y-coordinate
(1, 0)
(0, 1)
(-1, 0)
(0, 1)
90
1180cos
Recall, cosine is the x-coordinate
(1, 0)
(0, 1)
(-1, 0)
(0, 1)
180
01
0360tan
Recall, tangent is y / x
(1, 0)
(0, 1)
(-1, 0)
(0, 1)
360
Answer the question:
2tancos5
2sin3
360tan180cos590sin3
01513
8
Ex. 3 Evaluate the following exactly (without your calculator):
Try this example on your own first.Then, check out the solution.
3tan
6cos
4sin 2
Convert to degrees (if you need to)
Recall, radians = 180
3tan
6cos
4sin 2
3
180tan
6
180cos
4
180sin 2
3tan
6cos
4sin 2
3
180tan
6
180cos
4
180sin 2
60tan30cos45sin 2
Note that sin 2 = (sin ) 2
Evaluate the special angles
using the unit circle
(1, 0)
(0, 1)
(-1, 0)
(0, 1)
2
1,
2
330
6
2
2,
2
245
4
2
3,
2
160
3
These special angles can be read off the unit circle directly.
Answer the question:
3tan
6cos
4sin 2
60tan30cos45sin 2
21
23
2
3
2
22
1
3
2
3
4
2
21
23
2
3
2
22
2
3
2
1 1
1
3
2
3
4
2
21
23
2
3
2
22
Ex. 4 Express the following as a function of its related acute
angle and then evaluate:
Try this example on your own first.Then, check out the solution.
120sin
First, sketch the angle
y
xstart 0
90
180
120
Next, find the reference the angle
y
x60
120
The reference angle is 60
Using CAST, determine whether the trig ratio is
positive or negative
y
x60
Since the angle is in quadrant 2, sine is positive.
AS
T C
Express the trig ratio in terms of the reference angle:
120sin
60sin
Use the unit circle
to evaluate the special
angle exactly:
120sin
60sin
(1, 0)
(0, 1)
(-1, 0)
(0, 1)
2
3,
2
160
3
2
3
Recall, sine is the y-coordinate.
Ex. 5 Express the following as a function of its related acute
angle and then evaluate:
Try this example on your own first.Then, check out the solution.
6
11cot
First, convert to degrees and a primary trig ratio:
6
11cot
6
18011cot
330cot
Recall, radians = 180
6
11cot
6
18011cot
330cot
330tan
1
Next, sketch the angle
y
xstart 0, 360
90
180330
270
Next, find the reference the angle
The reference angle is 30
y
x30
330
Using CAST, determine whether the trig ratio is
positive or negative
Since the angle is in quadrant 4, tangent is negative (only cosine is positive)
C
y
x30
AS
T
Express the trig ratio in terms of the reference angle:
6
11cot
330tan
1
30tan
1
Use the unit circle
to evaluate the special
angle exactly:
Recall, tangent is y / x.
(1, 0)
(0, 1)
(-1, 0)
(0, 1)
2
1,
2
330
6
232
130tan
232
130tan
3
1
Answer the question:
6
11cot
30tan
1
311
3