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LESSON 5 Section 6.3
Trig Functions of Real Numbers
UNIT CIRCLE
Remember, the sine of a real number t (a number that corresponds to radians) is the y value of a point on a unit circle and the cosine of that real number is the x value of the point on a unit circle.
APPENDIX IV of your textbook shows a good unit circle.
Make a table of x and y values for the equation y = sin x.
x y x y
0 0 7π/6 -0.5
π/6 0.5 5π/4 -0.707
π/4 0.707 4π/3 -0.866
π/3 0.866 -3π/2 -1
π/2 1 5π/3 -0.866
2π/3 0.866 7π/4 -0.707
3π/4 0.707 11π/6 -0.5
5π/6 0.5 2π 0
π 0 13π/6 0.5
x y x y
2π 0 19π/6 -0.5
13π/6 0.5 13π/4 -0.707
9π/4 0.707 10π/3 -0.866
7π/3 0.866 7π/2 -1
5π/2 1 11π/3 -0.866
8π/3 0.866 15π/4 -0.707
11π/4 0.707 23π/6 -0.5
17π/6 0.5 4π 0
3π 0 25π/6 0.5
This is a second revolution around the unit circle. This is another ‘period’ of the curve.
y = sin x
• This is a periodic function. The period is 2π.
• The domain of the function is all real numbers.
• The range of the function is [-1, 1].
• It is a continuous function. The graph is shown on the next slide.
Graphing the sine curve for -2π ≤ x ≤ 2π.
-1
0
1
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
sin( )
: ,
: 1,1
: 2
y x
Domain
Range
Period
(0, 0)
(π/2, 1)
(π, 0)
(3π/2, - 1)
(2π, 0)
UNIT CIRCLE
Remember, the sine of a real number t (a number that corresponds to radians) is the y value of a point on a unit circle and the cosine of that real number is the x value of the point on a unit circle.
Make a table of x and y values for y = cos x
x y x y
0 1 7π/6 -0.866
π/6 0.866 5π/4 -0.707
π/4 0.707 4π/3 -0.5
π/3 0.5 -3π/2 0
π/2 0 5π/3 0.5
2π/3 -0.5 7π/4 0.707
3π/4 -0.707 11π/6 0.866
5π/6 -0.866 2π 1
π -1 13π/6 0.866
Remember, the y value in this table is actually the x value on the unit circle.
y = cos x
• This is a periodic function. The period is 2π.
• The domain of the function is all real numbers.
• The range of the function is [-1, 1].
• It is a continuous function. The graph is shown on the next slide.
Graphing the cosine curve for -2π ≤ x ≤ 2π.
-1
0
1
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
cos( )
: ,
: 1,1
: 2
y x
Domain
Range
Period
(0, 1)
(π/2, 0)
(π, - 1)
(3π/2, 0)
(2π, 1)
How do the graphs of the sine function and the cosine function
compare?• They are basically the same ‘shape’.
• They have the same domain and range.
• They have the same period.
• If you begin at –π/2 on the cosine curve, you have the sine curve.
sin cos( )
2y x x
, sin ___6
, sin _____6
x x
x x
The notation above is interpreted as: ‘as x approaches the number π/6 from the right (from values of x larger than π/6), what function value is sin x approaching?’ Since the sine curve is continuous (no breaks or jumps), the answer will be equal to exactly the sin (π/6) or ½ .
The notation below is interpreted as: ‘as x approaches the number π/6 from the left (from values of x smaller than π/6), what function value is sin x approaching?’ Again, since the sine curve is continuous, the answer will be equal to exactly the sin (π/6) or ½ .
Answer the following.
As ,cos _____
As , sin _____4
x x
x x
Find all the values x in the interval [0, 2) that satisfy the equation.
Use the graph to verify these values.
1cos
2x
-1
0
1
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Find all the values x in the interval [0, 2) that satisfy the equation.
1cos
2x
-1
0
1
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Q I Q IV
Find all the values x in the interval [0, 2) that satisfy the equation.
1cos
25
,3 3
x
x
-1
0
1
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Find all the values x in the interval [0, 2) that satisfy the equation.
Use the graph to verify these values.
1sin
2x
-1
0
1
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Find all the values x in the interval [0, 2) that satisfy the equation.
1sin
2x
-1
0
1
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Q III Q IV
Find all the values x in the interval [0, 2) that satisfy the equation.
1sin
25 7
,4 4
x
x
-1
0
1
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Make a table of x and y values for y = tan x
x y x y
-π/2 undefined 0.49π 31.821
-0.49π -31.821 π/2 undefined
-π/3 -1.732 0.51π -31.821
-π/4 -1 2π/3 -1.732
-π/6 -0.577 3π/4 -1
0 0 5π/6 -0.577
π/6 0.577 π 0
π/4 1 7π/6 0.577
π/3 1.732 5π/4 1
Remember, tan x is (sinx / cosx).
y = tan x• This is a periodic function. The period
is π.• The domain of the function is all real numbers,
except those of the form
π/2 +nπ.• The range of the function is all real numbers.• It is not a continuous function. The function is
undefined at -3π/2, -π/2, π/2, 3π/2, etc. There are vertical asymptotes at these values. The graph is shown on the next slide.
Graphing the tangent curve for -2π ≤ x ≤ 2π.
-10
-8
-6
-4
-2
0
2
4
6
8
10
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
tan
:{ | / 2 where n is an integer}
: ,
:
y x
Domain x x n
Range
Period
(-π/4, -1)
(π/4, 1)
As , tan _____2
As , tan _____2
x x
x x
For all x values where the tangent curve is continuous, approaching from the left or the right will equal the value of the tangent at x. However, the two cases above are different; because there is a vertical asymptote when x = -π/2. If approaching from the left (the smaller side), the answer is infinity. If approaching from the right (the larger side), the answer is negative infinity.
Find the answers.
As , tan _____
As , tan _____2
As , tan _____2
x x
x x
x x
Find all the values x in the interval [0, 2) that satisfy the equation.
tan x = 1
-10
-8
-6
-4
-2
0
2
4
6
8
10
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Q I Q III
Find all the values x in the interval [0, 2) that satisfy the equation.
-10
-8
-6
-4
-2
0
2
4
6
8
10
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
tan 1
5,
4 4
x
x
Find all the values x in the interval
that satisfy the equation.3
,2 2
1tan
3x
-10
-8
-6
-4
-2
0
2
4
6
8
10
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Find all the values x in the interval
that satisfy the equation.3
,2 2
1tan
3x
-10
-8
-6
-4
-2
0
2
4
6
8
10
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Q I Q III
Find all the values x in the interval
that satisfy the equation.3
,2 2
1tan
37
,6 6
x
x
-10
-8
-6
-4
-2
0
2
4
6
8
10
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Find all the values x in the interval
that satisfy the equation.3
,2 2
tan 1x
-10
-8
-6
-4
-2
0
2
4
6
8
10
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Find all the values x in the interval
that satisfy the equation.3
,2 2
tan 1x
-10
-8
-6
-4
-2
0
2
4
6
8
10
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Q IV Q II
Find all the values x in the interval
that satisfy the equation.3
,2 2
tan 1
3,
4 4
x
x
-10
-8
-6
-4
-2
0
2
4
6
8
10
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Sketch the graph of y = sin x + 1
This will be a graph of the basic sine function, but shifted one unit up.
The domain will be all real numbers. What would be the range?
Since the range of a basic sine function is [-1, 1], the domain of the function above would be [0, 2].
Sketch the graph of y = sin x + 1
-1
1
3
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Sketch the graph of y = cos x - 2
This would be the graph of a basic cosine function shifted 2 units down.
The domain is still all real numbers. What is the range?
The basic cosine function has a range of [-1, 1]. The range of the function above would be [-3, -1].
Sketch the graph of y = cos x - 2
-4
-3
-2
-1
0
1
-π 2
-π-3π 2
-2π π2
π 3π 2
2π
Find the intervals from –2π to 2π where the graph of y = tan x is:
a) Increasing
b) Decreasing
Remember: No brackets should be used on values of x where the function is not defined.
a) Increasing: [-2π, -3π/2) b) The function never decreases.
(-3π/2, -π/2)
(-π/2, π/2)
(π/2, 3π/2)
(3π/2, 2π]