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# 4.6 graphs of trig functions and inverse trig functions

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• 1. Graphs of Trig. Functions
• 2. Graphs of Trig. Functions
The graph of y=sin(x)
• 3. Graphs of Trig. Functions
The graph of y=sin(x)
• 4. Graphs of Trig. Functions
The graph of y=sin(x)
• 5. Graphs of Trig. Functions
The graph of y=sin(x)
• 6. Graphs of Trig. Functions
The graph of y=sin(x)
• 7. Graphs of Trig. Functions
The graph of y=sin(x)
• 8. Graphs of Trig. Functions
The graph of y=sin(x)
• 9. Graphs of Trig. Functions
The graph of y=sin(x)
• 10. Graphs of Trig. Functions
The graph of y=cos(x)
• 11. Graphs of Trig. Functions
The graph of y=cos(x)
• 12. Graphs of Trig. Functions
The graph of y=cos(x)
90o
180o
0o
• 13. Graphs of Trig. Functions
The graph of y=cos(x)
90o
180o
270o
360o
0o
• 14. Graphs of Trig. Functions
The graph of y=cos(x)
• 15. Graphs of Trig. Functions
The graph of y=cos(x)
The graph of y=sin(x)
• 16. Periodic Functions
• 17. Periodic Functions
Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
f(x) = f(x+b) for all x.
• 18. Periodic Functions
Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
f(x) = f(x+b) for all x.
The graph of a periodic function:
 Frank Ma
2006
p
• 19. Periodic Functions
Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
f(x) = f(x+b) for all x.
The smallest number p>0 such that f(x) = f(x+p)
is called the period of f(x).
The graph of a periodic function:
 Frank Ma
2006
p
• 20. Periodic Functions
Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
f(x) = f(x+b) for all x.
The smallest number p>0 such that f(x) = f(x+p)
is called the period of f(x).
The graph of a periodic function:
 Frank Ma
2006
p
one period
• 21. Periodic Functions
Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
f(x) = f(x+b) for all x.
The smallest number p>0 such that f(x) = f(x+p)
is called the period of f(x).
Over every interval of length p, the graph of a periodic function repeats itself.
The graph of a periodic function:
 Frank Ma
2006
p
one period
• 22. Periodic Functions
Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
f(x) = f(x+b) for all x.
The smallest number p>0 such that f(x) = f(x+p)
is called the period of f(x).
Over every interval of length p, the graph of a periodic function repeats itself.
The graph of a periodic function:
 Frank Ma
2006
p
x
x+p
p
one period
• 23. Periodic Functions
Given a function f(x), f(x) is said to be periodic if there exists a nonzero number b such that
f(x) = f(x+b) for all x.
The smallest number p>0 such that f(x) = f(x+p)
is called the period of f(x).
Over every interval of length p, the graph of a periodic function repeats itself.
The graph of a periodic function:
 Frank Ma
2006
p
x
x+p
p
one period
f(x) = f(x+p) for all x
• 24. Periodic Functions
sin(x) and cos(x) are periodic with period p=2π.
• 25. Periodic Functions
sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
For y=cos(x):
• 26. Periodic Functions
sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
For y=cos(x):
• 27. Periodic Functions
sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
For y=cos(x):
• 28. Periodic Functions
sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
For y=cos(x):
• 29. Periodic Functions
sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
For y=cos(x):
• 30. Periodic Functions
sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
For y=sin(x):
0
• 31. Periodic Functions
sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
For y=sin(x):
0
• 32. Periodic Functions
sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
For y=sin(x):
0
• 33. Periodic Functions
sin(x) and cos(x) are periodic with period p=2π. The graphs of y=sin(x) and y=cos(x) repeats itself every 2π period.
For y=sin(x):
0
• 34. Periodic Functions
The basic period for:
y=sin(x)
• 35. Periodic Functions
The basic period for:
y=sin(x) y=cos(x)
1
-1
• 36. Periodic Functions
The basic period for:
y=sin(x) y=cos(x)
1
-1
The Graph of Tangent
• 37. Periodic Functions
The basic period for:
y=sin(x) y=cos(x)
1
-1
The Graph of Tangent
The function tan(x) is not defined when cos(x) is 0,
i.e. when x = ±π/2, ±3π/2, ±5π/2, ..
• 38. Periodic Functions
The basic period for:
y=sin(x) y=cos(x)
1
-1
The Graph of Tangent
The function tan(x) is not defined when cos(x) is 0,
i.e. when x = ±π/2, ±3π/2, ±5π/2, ..
 Frank Ma
2006
As the values of x goes from 0 to π/2 the values of sin(x) goes from 0 to 1,
• 39. Periodic Functions
The basic period for:
y=sin(x) y=cos(x)
1
-1
The Graph of Tangent
The function tan(x) is not defined when cos(x) is 0,
i.e. when x = ±π/2, ±3π/2, ±5π/2, ..
 Frank Ma
2006
As the values of x goes from 0 to π/2 the values of sin(x) goes from 0 to 1, but the values of cos(x) goes from 1 to 0. So tan(x) goes from 0 to +∞.
• 40. The Graph of Tangent
Since tan(x) is odd, so as the values of x goes from
0 to -π/2 we get the corresonding negative outputs
for tan(x).
• 41. The Graph of Tangent
Since tan(x) is odd, so as the values of x goes from
0 to -π/2 we get the corresonding negative outputs
for tan(x). Specifically,
x
π/6
0
π/4
π/3
-π/2
-π/6
-π/4
-π/3
π/2
tan(x)
• 42. The Graph of Tangent
Since tan(x) is odd, so as the values of x goes from
0 to -π/2 we get the corresonding negative outputs
for tan(x). Specifically,
x
π/6
0
π/4
π/3
-π/2
-π/6
-π/4
-π/3
π/2

0
1/3
1
3
tan(x)
• 43. The Graph of Tangent
Since tan(x) is odd, so as the values of x goes from
0 to -π/2 we get the corresonding negative outputs
for tan(x). Specifically,
x
π/6
0
π/4
π/3
-π/2
-π/6
-π/4
-π/3
π/2

0
1/3
1
3
-1/3
-1
-3
-∞
tan(x)
• 44. The Graph of Tangent
Since tan(x) is odd, so as the values of x goes from
0 to -π/2 we get the corresonding negative outputs
for tan(x). Specifically,
x
π/6
0
π/4
π/3
-π/2
-π/6
-π/4
-π/3
π/2

0
1/3
1
3
-1/3
-1
-3
-∞
tan(x)
0
π/2
-π/2
• 45. The Graph of Tangent
Since tan(x) is odd, so as the values of x goes from
0 to -π/2 we get the corresonding negative outputs
for tan(x). Specifically,
x
π/6
0
π/4
π/3
-π/2
-π/6
-π/4
-π/3
π/2

0
1/3
1
3
-1/3
-1
-3
-∞
tan(x)
The same pattern repeats itself every πinterval.
0
π/2
-π/2
• 46. The Graph of Tangent
Since tan(x) is odd, so as the values of x goes from
0 to -π/2 we get the corresonding negative outputs
for tan(x). Specifically,
x
π/6
0
π/4
π/3
-π/2
-π/6
-π/4
-π/3
π/2

0
1/3
1
3
-1/3
-1
-3
-∞
tan(x)
The same pattern repeats itself every πinterval.
In other words,
y = tan(x) is a periodic function with period πas shown in the graph.
0
π/2
-π/2
• 47. The Graph of Tangent
Since tan(x) is odd, so as the values of x goes from
0 to -π/2 we get the corresonding negative outputs
for tan(x). Specifically,
x
π/6
0
π/4
π/3
-π/2
-π/6
-π/4
-π/3
π/2

0
1/3
1
3
-1/3
-1
-3
-∞
tan(x)
The same pattern repeats itself every πinterval.
In other words,
y = tan(x) is a periodic function with period πas shown in the graph.
π

0
π/2
-π/2
3π/2
-3π/2
y = tan(x)
• 48. The Graph of Inverse Trig-Functions
Recalll that for y = cos-1(x), then 0 < y < π.
• 49. The Graph of Inverse Trig-Functions
Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.
• 50. The Graph of Inverse Trig-Functions
Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.
(-1, π)
(0, π/2)
(1, 0)
-1
1
The graph of y = cos-1(x)
• 51. The Graph of Inverse Trig-Functions
Recalll that for y = cos-1(x), then 0 < y < π. It's graph may be plotted in the similar manner.
(-1, π)
(0, π/2)
(1, 0)
-1
1
The graph of y = cos-1(x)
Remark: The above graphs of y = sin-1(x) and
y = cos-1(x) are the complete graphs (i.e. that's all there is).
• 52. The Graph of Inverse Trig-Functions
The domain of y = tan-1(x) is all real numbers and
the output y is restricted to -π/2 < y < π.
• 53. The Graph of Inverse Trig-Functions
The domain of y = tan-1(x) is all real numbers and
the output y is restricted to -π/2 < y < π.
x

0
1/3
1
3
-1/3
-1
-3
-∞
tan-1(x)
• 54. The Graph of Inverse Trig-Functions
The domain of y = tan-1(x) is all real numbers and
the output y is restricted to -π/2 < y < π.
x

0
1/3
1
3
-1/3
-1
-3
-∞
tan-1(x)
π/6
0
π/4
π/3
π/2
• 55. The Graph of Inverse Trig-Functions
The domain of y = tan-1(x) is all real numbers and
the output y is restricted to -π/2 < y < π.
x

0
1/3
1
3
-1/3
-1
-3
-∞
tan-1(x)
π/6
0
π/4
π/3
-π/2
-π/6
-π/4
-π/3
π/2
• 56. The Graph of Inverse Trig-Functions
The domain of y = tan-1(x) is all real numbers and
the output y is restricted to -π/2 < y < π.
x

0
1/3
1
3
-1/3
-1
-3
-∞
tan-1(x)
π/6
0
π/4
π/3
-π/2
-π/6
-π/4
-π/3
π/2
Here is the graph of y = tan-1(x)
y = π/2
(1,π/4)
(0,0)
(-1,-π/4)
y = -π/2
• 57. The Graph of Inverse Trig-Functions
The domain of y = tan-1(x) is all real numbers and
the output y is restricted to -π/2 < y < π.
x

0
1/3
1
3
-1/3
-1
-3
-∞
tan-1(x)
π/6
0
π/4
π/3
-π/2
-π/6
-π/4
-π/3
π/2
Here is the graph of y = tan-1(x)
y = π/2
(1,π/4)
(0,0)
(-1,-π/4)
y = -π/2
Remark: y =tan-1(x) has two horizontal asymptoes.