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1
Trigonometry
Trigonometry begins in the right triangle, but it doesn’t have to be restricted to triangles. The trigonometric functions carry the ideas of triangle trigonometry into a broader world of real-valued functions and wave forms.
2
Trigonometry Topics
Radian Measure The Unit Circle Trigonometric Functions Larger Angles Graphs of the Trig Functions Trigonometric Identities Solving Trig Equations
3
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 the radius of the circle.
rs sr
4
Radian Measure
degrees
360
radians
2
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
5
Sample Problems Find the degree
measure equivalent of radians.
degrees
360
radians
210
360
r
2
2360 420
420
360
7
6
r
r
degrees
360
radians
360
3 4
2
22 270
135
d
d
d
3
4
Find the radian measure equivalent of 210°
6
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 circle somewhere in quadrant I.
7
The Unit Circle
Connect the origin to the point, and from that point drop a perpendicular to the x-axis.
This creates a right triangle with hypotenuse of 1.
8
The Unit Circle
The length of its legs are the x- and y-coordinates of the chosen point.
Applying the definitions of the trigonometric ratios to this triangle gives
x
y1
is the angle of rotation
xx
1
cosyy
1
sin
9
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 .
The other trigonometric functions can be defined from these.
yy
1
)sin(
xx
1
)cos(
10
Trigonometric Functions
sin( ) y
x
y1
is the angle of rotation
x
y)tan(
y
1)csc(
x)cos(x
1)sec(
y
x)cot(
11
Around the Circle
As that point moves around the unit circle into quadrants II, III, and IV, the new definitions of the trigonometric functions still hold.
12
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.
The acute angle which produces the same values is called the reference angle.
13
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 dropping a perpendicular from the point on the unit circle to the x-axis.
14
Quadrant II
Original angle
Reference angle
For an angle, , in quadrant II, the reference angle is
In quadrant II, sin() is positive cos() is negative tan() is negative
15
Quadrant III
Original angle
Reference angle
For an angle, , in quadrant III, the reference angle is
- In quadrant III,
sin() is negative cos() is negative tan() is positive
16
Quadrant IV
Original angle
Reference angle
For an angle, , in quadrant IV, the reference angle is 2
In quadrant IV, sin() is negative cos() is positive tan() is negative
17
All Seniors Take Calculus Use the phrase “All Seniors Take Calculus”
to remember the signs of the trig functions in different quadrants.
AllSeniors
Take Calculus
All functions are positive
Sine is positive
Tan is positive Cos is positive
18
Angles measured in degrees:
1sin 45 cos45 and tan 45 1
2
Angles measured in radians:
1sin / 4 cos / 4 and tan / 4 1
2
Special Right Triangles
19
Angles measured in degrees:
1sin30 cos60
2
3sin 60 cos30
21
tan 60 3tan30
Special Right Triangles
20
The 16-Point Unit Circle
0
1/2
2 /2
3 /2
1
1
3 /2
2 /2
1/2
0
0
3 /3
1
3
2
2
2 3 /3
1
1
2 3 /3
2
2
3
1
3 /3
0
3 /2
2 /2
1/2
0
1/2
2 /2
3 /2
1
3
1
3 /3
0
2 3 /3
2
2
2
2
2 3 /3
1
3 /3
1
3
1/2
2 /2
3 /2
1
3 /2
2 /2
1/2
0
3 /2
2 /2
1/2
0
1/2
2 /2
3 /2
1
3 /3
1
3
3
1
3 /3
0
2
2
2 3 /3
1
2 3 /3
2
2
2 3 /3
2
2
2
2
2 3 /3
1
3
1
3 /3
0
3 /3
1
3
22
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.
Graphs of the Trig Functions
23
The graph of is 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
patterns that are completed in a space of 2. The period of the function is The two remaining numbers, h and k, tell the
translation of the wave from the origin.
Graphs of the Trig Functions
khxbay )(sin
2b
24
Sample Problem Which of the following
equations best describes the graph shown? (A) y = 3sin(2x) - 1 (B) y = 2sin(4x) (C) y = 2sin(2x) - 1 (D) y = 4sin(2x) - 1 (E) y = 3sin(4x)
-2p -1p 1p 2p
5
4
3
2
1
-1
-2
-3
-4
-5
25
Sample Problem Find the baseline between
the high and low points. Graph is translated -1
vertically. Find height of each peak.
Amplitude is 3 Count number of waves in
2 Frequency is 2
-2p -1p 1p 2p
5
4
3
2
1
-1
-2
-3
-4
-5
y = 3sin(2x) - 1
26
Cosine The graph of y = cos(x) resembles the graph
of y = sin(x) but is shifted, or translated, units to the left.
It fluctuates from 1
to 0, down to –1,
back to 0 and up to
1, in a space of 2.
Graphs of the Trig Functions
27
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.
Amplitude a Height of each peakFrequency b Number of full wave patterns Period 2/b Space required to complete waveTranslation h, k Horizontal and vertical shift
khxbay )(cos
28
Which of the following equations best describes the graph? (A) y = 3cos(5x) + 4 (B) y = 3cos(4x) + 5 (C) y = 4cos(3x) + 5 (D) y = 5cos(3x) + 4 (E) y = 5sin(4x) + 3
Sample Problem
-2p -1p 1p 2p
8
6
4
2
29
Find the baseline Vertical translation + 4
Find the height of peak Amplitude = 5
Number of waves in 2 Frequency =3
Sample Problem
-2p -1p 1p 2p
8
6
4
2
y = 5 cos(3x) + 4
30
Tangent The tangent function has a
discontinuous graph, repeating in a period of .
Cotangent Like the tangent, cotangent is
discontinuous. Discontinuities of the
cotangent are units left of those for tangent.
Graphs of the Trig Functions
2
31
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 troughs of its reciprocal function.
32
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.
The most important ones should be committed to memory.
33
Trigonometric Identities
Reciprocal Identities
xx
sec
1cos
tansin
cosx
x
x
cotcos
sinx
x
x
Quotient Identities
xx
cot
1tan
xx
csc
1sin
34
Trigonometric Identities
Cofunction Identities The function of an angle = the
cofunction of its complement.
)90cot(tan xx
)90csc(sec xx
)90cos(sin xx
35
Trigonometric Identities
Pythagorean Identities
The fundamental
Pythagorean identity
Divide the first by sin2x
Divide the first by cos2x xx 22 sec1tan
xx 22 csccot1
1cossin 22 xx
36
Trigonometric Identities
2 2
2
2
2
cos2 cos sin
cos2 1 2sin
cos2 2cos 1
sin 2 2sin cos
2 tantan 2
1 tan
2sin cos sin( ) sin( )
2cos cos cos( ) cos( )
2sin sin cos( ) cos( )
37
Trigonometric Identitiescos( ) cos cos sin sin
cos( ) cos cos sin sin
sin( ) sin cos cos sin
sin( ) sin cos cos sin
tan tantan( )
1 tan tan
tan tantan( )
1 tan tan
-
38
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 Solve the equation for this new variable Reinsert the trig function Determine the argument which will produce the
desired value
39
Solving Trig Equations
To solving trig equations:
Use identities to simplify
Let variable = trig function
Solve for new variable
Reinsert the trig function
Determine the argument
40
Sample Problem
Solve
90or150,30 xx
3 3 2 02 sin cosx x
1sinor2
1sin xx
0cos2sin33 2 xx
0)sin1)(sin21( xx
0sin2sin31 2 xx
0)sin1(2sin33 2 xx
41
All these relationships are based on the assumption that the triangle is a right triangle.
It is possible, however, to use trigonometry to solve for unknown sides or angles in non-right triangles.
Law of Sines and Cosines
42
In geometry, you learned that the largest angle of a triangle was opposite the longest side, and the smallest angle opposite the shortest side.
The Law of Sines says that the ratio of a side to the sine of the opposite angle is constant throughout the triangle.
a
A
b
B
c
Csin( ) sin( ) sin( )
Law of Sines
43
In ABC, mA = 38, mB = 42, and BC = 12 cm. Find the length of side AC. Draw a diagram to see the position of the given
angles and side. BC is opposite A You must find AC, the side opposite B.
A B
C
Law of Sines
44
.... Find the length of side AC. Use the Law of Sines with mA = 38, mB = 42, and
BC = 12
a
A
b
Bsin( ) sin( )
12
38 42sin( ) sin( ) b
38sin42sin12 b
38sin
42sin12b
041.13042.13
029.8
Law of Sines
45
WarningWarning
The Law of Sines is useful when you know the sizes of two sides and one angle or two angles and one side.
However, the results can be ambiguous if the given information is two sides and an angle other than the included angle (ssa).
46
Law of Cosines
If you apply the Law of Cosines to a right triangle, that extra term becomes zero, leaving just the Pythagorean Theorem.
The Law of Cosines is most useful when you know the lengths of all three sides and need to
find an angle, or when you two sides and the included angle.
47
Triangle XYZ has sides of lengths 15, 22, and 35. Find the measure of the angle C.
c a b ab C
C
C
C
2 2 2
2 2 2
2
35 15 22 2 15 22
1225 225 484 660
1225 709 660
cos( )
cos( )
cos( )
cos( )
15 22
35
C
Law of Cosines
48
... Find the measure of the largest angle of the triangle.
516 660
516
6607818
7818 14141
cos( )
cos( ) .
cos ( . ) .
C
C
C
15 22
35
Law of Cosines
49
Laws of Sines and Cosines
a
b
c
B
C
A
Cabbac
C
c
B
b
A
a
cos2
sinsinsin
222
Law of Sines:
Law of Cosines: