Contributed by:
Topics:
1. Radian Measure
2. The Unit Circle
3. Trigonometric Functions
4. Larger Angles
5. Graphs of the Trig Functions
6. Trigonometric Identities
7. Solving Trig Equations
1.
Preparing for the SAT II
Trigonometry
2.
Trigonometry
Trigonometry begins in the right
triangle, but it doesn’t have to be
restricted to triangles. The
trigonometric functions carry the
©Carolyn C. Wheater, 2000
ideas of triangle trigonometry into a
broader world of real-valued
functions and wave forms.
2
3.
Trigonometry Topics
Radian Measure
The Unit Circle
Trigonometric Functions
Larger Angles
©Carolyn C. Wheater, 2000
Graphs of the Trig Functions
Trigonometric Identities
Solving Trig Equations
3
4.
Radian Measure
To talk about trigonometric functions, it is
helpful to move to a different system of
angle measure, called radian measure.
A radian is the measure of a central angle
whose intercepted arc is equal in length to
©Carolyn C. Wheater, 2000
the radius of the circle.
4
5.
Radian Measure
There are 2 radians in a full rotation --
once around the circle
There are 360° in a full rotation
To convert from degrees to radians or
radians to degrees, use the proportion
©Carolyn C. Wheater, 2000
degrees radians
o
=
360 2π
5
6.
Sample Problems
Find the degree Find the radian
measure equivalent measure equivalent
3π
of radians. of 210°
4
degrees radians degrees radians
o
= o
=
360 2π 360 2π
©Carolyn C. Wheater, 2000
d 3π 4 210o r
= o
=
360o
2π 360 2π
2πd = 270π 360r = 420π
420π 7π
d = 135o r= = 6
360 6
7.
The Unit Circle
Imagine a circle on the
coordinate plane, with its
center at the origin, and
a radius of 1.
Choose a point on the
©Carolyn C. Wheater, 2000
circle somewhere in
quadrant I.
7
8.
The Unit Circle
Connect the origin to the
point, and from that point
drop a perpendicular to
the x-axis.
This creates a right
©Carolyn C. Wheater, 2000
triangle with hypotenuse
of 1.
8
9.
The Unit Circle
is the
The length of its legs are angle of
rotation
the x- and y-coordinates of
the chosen point.
Applying the definitions of 1
y
the trigonometric ratios to x
©Carolyn C. Wheater, 2000
this triangle gives
bg
cos θ = =x
x
1
y
sin(θ ) = = y
1
9
10.
The Unit Circle
The coordinates of the chosen point are the
cosine and sine of the angle .
This provides a way to define functions sin()
and cos() for all real numbers .
y bg x
cos θ = =x
©Carolyn C. Wheater, 2000
sin(θ ) = = y
1 1
The other trigonometric functions can be
defined from these.
10
11.
Trigonometric Functions
is the
sin(θ ) = y bg
csc θ =
1
y
angle of
rotation
bg
cos θ =x bg
sec θ =
1
x
1
y
x
©Carolyn C. Wheater, 2000
bg
tan θ =
y
x
bg
cot θ =
x
y
11
12.
Around the Circle
As that point
moves around the
unit circle into
quadrants II, III,
and IV, the new
©Carolyn C. Wheater, 2000
definitions of the
trigonometric
functions still hold.
12
13.
Reference Angles
The angles whose terminal sides fall in
quadrants II, III, and IV will have values of
sine, cosine and other trig functions which
are identical (except for sign) to the values
of angles in quadrant I.
©Carolyn C. Wheater, 2000
The acute angle which produces the same
values is called the reference angle.
13
14.
Reference Angles
The reference angle is the angle between
the terminal side and the nearest arm of the
x-axis.
The reference angle is the angle, with vertex
at the origin, in the right triangle created by
©Carolyn C. Wheater, 2000
dropping a perpendicular from the point on
the unit circle to the x-axis.
14
15.
Quadrant II
Original angle For an angle, , in
quadrant II, the
reference angle is
In quadrant II,
©Carolyn C. Wheater, 2000
Reference angle sin() is positive
cos() is negative
tan() is negative
15
16.
Quadrant III
Original angle For an angle, , in
quadrant III, the
reference angle is
-
In quadrant III,
©Carolyn C. Wheater, 2000
Reference angle sin() is negative
cos() is negative
tan() is positive
16
17.
Quadrant IV
For an angle, , in
Reference angle quadrant IV, the
reference angle is
2
In quadrant IV,
©Carolyn C. Wheater, 2000
sin() is negative
cos() is positive
Original angle tan() is negative
17
18.
All Star Trig Class
Use the phrase “All Star Trig Class” to
remember the signs of the trig functions in
different quadrants.
Star All
Sine is positive All functions
©Carolyn C. Wheater, 2000
are positive
Trig Class
Tan is positive Cos is positive
18
19.
Graphs of the Trig Functions
Sine
The most fundamental sine wave, y=sin(x),
has the graph shown.
It fluctuates from 0 to a high of 1, down to –1,
and back to 0, in a space of 2.
©Carolyn C. Wheater, 2000
19
20.
Graphs of the Trig Functions
The graph of cb gh
y = a sinb x −h +is
k determined
by four numbers, a, b, h, and k.
The amplitude, a, tells the height of each peak and
the depth of each trough.
The frequency, b, tells the number of full wave
©Carolyn C. Wheater, 2000
patterns that are completed in a space of 2.
The period of the function is 2π
b
The two remaining numbers, h and k, tell the
translation of the wave from the origin.
20
21.
Sample Problem
Which of the following
equations best describes
the graph shown?
(A) y = 3sin(2x) - 1
(B) y = 2sin(4x)
©Carolyn C. Wheater, 2000
(C) y = 2sin(2x) - 1
(D) y = 4sin(2x) - 1
(E) y = 3sin(4x)
21
22.
Sample Problem
Find the baseline between the
high and low points.
Graph is translated -1
vertically.
Find height of each peak.
Amplitude is 3
©Carolyn C. Wheater, 2000
Count number of waves in 2
Frequency is 2
y = 3sin(2x) - 1
22
23.
Graphs of the Trig Functions
Cosine
The graph of y=cos(x) resembles the graph of
y=sin(x) but is shifted, or translated, π units to
2
the left.
It fluctuates from 1
©Carolyn C. Wheater, 2000
to 0, down to –1,
back to 0 and up to
1, in a space of 2.
23
24.
Graphs of the Trig Functions
The values of a, b, h, and k change the shape
and location of the wave as for the sine.
cb gh
y = a cosb x −h + k
Amplitude a Height of each peak
©Carolyn C. Wheater, 2000
Frequency b Number of full wave patterns
Period 2/b Space required to complete wave
Translation h, k Horizontal and vertical shift
24
25.
Sample Problem
Which of the following
equations best describes
the graph?
(A) y = 3cos(5x) + 4
(B) y = 3cos(4x) + 5
©Carolyn C. Wheater, 2000
(C) y = 4cos(3x) + 5
(D) y = 5cos(3x) +4
(E) y = 5sin(4x) +3
25
26.
Sample Problem
Find the baseline
Vertical translation + 4
Find the height of peak
Amplitude = 5
Number of waves in
©Carolyn C. Wheater, 2000
2
Frequency =3 y = 5cos(3x) + 4
26
27.
Graphs of the Trig Functions
Tangent
The tangent function has a
discontinuous graph,
repeating in a period of .
Cotangent
©Carolyn C. Wheater, 2000
Like the tangent, cotangent is
discontinuous.
• Discontinuities of the cotangent
are units left
π of those for
2
tangent. 27
28.
Graphs of the Trig Functions
Secant and Cosecant
The secant and cosecant functions are the
reciprocals of the cosine and sine functions
respectively.
Imagine each graph is balancing on the peaks and
©Carolyn C. Wheater, 2000
troughs of its reciprocal function.
28
29.
Trigonometric Identities
An identity is an equation which is true for
all values of the variable.
There are many trig identities that are useful
in changing the appearance of an
expression.
©Carolyn C. Wheater, 2000
The most important ones should be
committed to memory.
29
30.
Trigonometric Identities
Reciprocal Identities Quotient Identities
1
sin x = sin x
csc x tan x =
cos x
1
cos x =
©Carolyn C. Wheater, 2000
sec x cos x
cot x =
sin x
1
tan x =
cot x
30
31.
Trigonometric Identities
Cofunction Identities
The function of an angle = the cofunction of its
complement.
o
sin x = cos(90 −x )
©Carolyn C. Wheater, 2000
o
sec x = csc(90 −x )
o
tan x = cot(90 −x )
31
32.
Trigonometric Identities
Pythagorean Identities
The fundamental Pythagorean identity
sin 2 x + cos2 x = 1
2 2
Divide the first by sin x 2 1 + cot x = csc x
©Carolyn C. Wheater, 2000
2 2
Divide the first by cos x 2 tan x + 1 = sec x
32
33.
Solving Trig Equations
Solve trigonometric equations by following
these steps:
If there is more than one trig function, use
identities to simplify
Let a variable represent the remaining function
©Carolyn C. Wheater, 2000
Solve the equation for this new variable
Reinsert the trig function
Determine the argument which will produce the
desired value
33
34.
Solving Trig Equations
To solving trig equations:
Use identities to simplify
Let variable = trig function
Solve for new variable
©Carolyn C. Wheater, 2000
Reinsert the trig function
Determine the argument
34
35.
Sample Problem
Solve 3 −3 sinx −2 cos2 x = 0
Use the Pythagorean
2
identity 3 −3 sin x −2 cos x=0
• (cos2x = 1 - sin2x)
Distribute
c h
3 −3 sinx −2 1 −sin2 x = 0
3 −3 sinx −2 + 2 sin2 x = 0
©Carolyn C. Wheater, 2000
Combine like terms
1 −3 sinx + 2 sin2 x = 0
Order terms
2 sin2 x −3 sinx + 1 = 0
35
36.
Sample Problem
Solve 3 −3 sinx −2 cos2 x = 0
Let t = sin x 2 sin 2 x −3 sin x + 1 = 0
2t 2 −3t + 1 = 0
Factor and solve. (2t −1)(t −1) = 0
2t −1 = 0 t −1 = 0
©Carolyn C. Wheater, 2000
2t = 1 t =1
1
t=
2
36
37.
Sample Problem
Solve 3 −3 sinx −2 cos2 x = 0
Replace t = sin x.
π 5π
t = sin(x) = ½ when x = or
6 6
π
t = sin(x) = 1 when x=
2
©Carolyn C. Wheater, 2000
π 5π π
So the solutions are x= , ,
6 6 2
37
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