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Semelhante a Trig overview (20)
Trig overview
- 2. ©CarolynC.Wheater,2000
2
Trigonometry
Trigonometry begins in the rightTrigonometry begins in the right
triangle, but it doesn’t have to betriangle, but it doesn’t have to be
restricted to triangles. Therestricted to triangles. The
trigonometric functions carry thetrigonometric functions carry the
ideas of triangle trigonometry into aideas of triangle trigonometry into a
broader world of real-valuedbroader world of real-valued
functions and wave forms.functions and wave forms.
- 4. ©CarolynC.Wheater,2000
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
the radius of the circle.
- 6. ©CarolynC.Wheater,2000
6
Sample Problems
Find the degree
measure equivalent
of radians.
degrees
360
radians
210
360
r
=
=
=
= =
2
2
360 420
420
360
7
6
π
π
π
π π
r
r
degrees
360
radians
360
3 4
=
=
=
=
2
2
2 270
135
π
π
π
π π
d
d
d
3
4
π
Find the radian
measure equivalent
of 210°
- 9. ©CarolynC.Wheater,2000
9
The Unit Circle
sin( )θ = =
y
y
1
cos θbg= =
x
x
1
x
y
1
θ is the
angle of
rotation
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
- 10. ©CarolynC.Wheater,2000
10
The Unit Circle
sin( )θ = =
y
y
1
cos θbg= =
x
x
1
The coordinates of the chosen point are the
cosine and sine of the angle θ.
This provides a way to define functions sin(θ)
and cos(θ) for all real numbers θ.
The other trigonometric functions can be
defined from these.
- 13. ©CarolynC.Wheater,2000
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.
The acute angle which produces the same
values is called the reference angle.
- 14. ©CarolynC.Wheater,2000
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
dropping a perpendicular from the point on
the unit circle to the x-axis.
- 18. ©CarolynC.Wheater,2000
18
All Star Trig Class
Use the phrase “All Star Trig Class” to
remember the signs of the trig functions in
different quadrants.
AllStar
Trig Class
All functions
are positive
Sine is positive
Tan is positive Cos is positive
- 19. ©CarolynC.Wheater,2000
19
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 2π.
Graphs of the Trig Functions
- 20. ©CarolynC.Wheater,2000
20
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 2π.
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
y a b x h k= − +sin b gc h
2π
b
- 21. ©CarolynC.Wheater,2000
21
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)
−2π −1π 1π 2π
5
4
3
2
1
−1
−2
−3
−4
−5
- 22. ©CarolynC.Wheater,2000
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
Count number of waves in
2π
Frequency is 2
−2π −1π 1π 2π
5
4
3
2
1
−1
−2
−3
−4
−5
y = 3sin(2x) - 1
- 23. ©CarolynC.Wheater,2000
23
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 2π.
Graphs of the Trig Functions
π
2
- 24. ©CarolynC.Wheater,2000
24
Graphs of the Trig Functions
y a b x h k= − +cos b gc h
Amplitude a Height of each peak
Frequency b Number of full wave patterns
Period 2π/b Space required to complete wave
Translation h, k Horizontal and vertical shift
The values of a, b, h, and k change the
shape and location of the wave as for the
sine.
- 25. ©CarolynC.Wheater,2000
25
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
−2π −1π 1π 2π
8
6
4
2
- 26. ©CarolynC.Wheater,2000
26
Find the baseline
Vertical translation + 4
Find the height of
peak
Amplitude = 5
Number of waves in
2π
Frequency =3
Sample Problem
−2π −1π 1π 2π
8
6
4
2
y = 5cos(3x) + 4
- 27. ©CarolynC.Wheater,2000
27
Tangent
The tangent function has a
discontinuous graph,
repeating in a period of π.
Cotangent
Like the tangent, cotangent is
discontinuous.
• Discontinuities of the
cotangent are units left of
those for tangent.
Graphs of the Trig Functions
π
2
- 28. ©CarolynC.Wheater,2000
28
Graphs of the Trig Functions
y=sec(x)
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.
- 33. ©CarolynC.Wheater,2000
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
Solve the equation for this new variable
Reinsert the trig function
Determine the argument which will produce the
desired value
- 35. ©CarolynC.Wheater,2000
35
Solve
Use the Pythagorean
identity
• (cos2
x = 1 - sin2
x)
Distribute
Combine like terms
Order terms
Sample Problem
3 3 2 0
3 3 2 1 0
3 3 2 2 0
1 3 2 0
2 3 1 0
2
2
2
2
2
− − =
− − − =
− − + =
− + =
− + =
sin cos
sin sin
sin sin
sin sin
sin sin
x x
x x
x x
x x
x x
c h
3 3 2 02
− − =sin cosx x
- 36. ©CarolynC.Wheater,2000
36
Let t = sin x
Factor and solve.
Sample Problem
Solve 3 3 2 02
− − =sin cosx x
2 3 1 02
sin sinx x− + =
2 3 1 0
2 1 1 0
2 1 0 1 0
2 1 1
1
2
2
t t
t t
t t
t t
t
− + =
− − =
− = − =
= =
=
( )( )
