Unformatted text preview: Pre – Calculus Math 40S: Explained!
www.math40s.com 67 TRIGONOMETRY LESSON SEVEN
Part I Graphing Radian Functions In application questions, when your xaxis is time, distance, or some other unit, you must
do your trig graph in radian mode. Basically, whenever you see integers on the xaxis,
(rather than degrees or radian fractions) you need to be in radian mode.
In the graph on the
right, notice how
the variable is time
instead of θ. This
is a clue you need
radian mode. y =1.23cos 2π
t +1.85
12.1 Example 1: Given the equation: y =13.2cos 2π
(t 101)+6.5 find
342 appropriate units for your window and graph with the TI83.
1) First you must figure out the period so you know how long a cycle is.
2π
2π
342
P er iod =
=
= 2π ×
= 342
2π
b
2π
342
The length of one complete cosine cycle is 342 units.
Also, since the pattern starts at 101 due to the phase shift, we want to see all the way to 443
(phase shift + period) for the complete picture.
2) Next you need to know where the minimum and maximum values are using the following
formulas:
Minimum = d — a
Maximum = d + a
Minimum = 6.5 — 13.2
Maximum = 6.5 + 13.2
Minimum = 6.7
Maximum = 19.7
3) Now choose a scale. The best way to do this is by picking numbers large enough that you
won’t have too many ticks on either axis. For the xaxis, we’re going from 0 to about 450,
so use a scale of 100. For the yaxis, since we want to see from 7 to +20, a scaling of 5 would
be good. Of course, these are just guidelines and you could use several different scales and
still obtain a good graph.
5) Now draw the graph:
4) Use the following
y
window settings:
20
Xmin: 0
Xmax: 443
Xscl: 20
Ymin: 6.7
Ymax:19.7
Yscl: 5 15
10
5
t
100 200 300 400 5 Pre – Calculus Math 40S: Explained!
www.math40s.com 68 TRIGONOMETRY LESSON SEVEN
Part I Graphing Radian Functions Questions: For each of the following equations, find appropriate window settings.
Then draw the graph using your TI83. 1. π y =17.2cos (t 7)+ 0.5
3 y = 2.2sin0.123π (t +1.7)+15.2 2. 2
0 2
0 1
5 1.
75 1
0 1
5 5 1.
25
1
0
2 4 6 8 1
0 1
2 5 75
. 0
1 5 5
1 25
. 0
2 3. 2 y = 20.1sin 2π
(t  265)+ 6.2
300 4. 4 6 8 1
0 1
2 1
4 y = 3.2cos0.18π t +17 2
0 3
0 1.
75
1
5 2
0 1.
25 1
0 1
0
75
. 10
0 20
0 30
0 40
0 50
0 60
0 5
25
. 0
1
2 4 6 8 1
0 1
2 Pre – Calculus Math 40S: Explained!
www.math40s.com 69 TRIGONOMETRY LESSON SEVEN
Part I Answers: 1.
1) Find the period: Period = Graphing Radian Functions 2π 2π
3
=
= 2π × = 6
π
b
π
3 Since the cosine pattern starts at 7 due to the phase shift, and the period is 6, we want to see up
to 13 on the xaxis.
2) Find the minimum and maximum:
Minimum = d — a = 0.5 — 17.2 = 16.7
Maximum = d + a = 0.5 + 17.2 = 17.7 5) Draw the graph. 3) Choose a scaling, 2 for the xaxis and
5 for the yaxis will work fine.
4) Use the following window settings:
Xmin: 0
Xmax: 13
Xscl: 2
Ymin: 16.7
Ymax:17.7
Yscl: 5 2.
1) Find the period: Period = 2π
2π
=
= 16.23
b 0.123π Since the sine pattern starts at 1.7 due to the phase shift, and the period is 16.23, we want to
see at least up to 14.53 on the xaxis for the complete picture.
2) Find the minimum and maximum:
Minimum = d — a = 15.2 — 2.2 = 13
Maximum = d + a = 15.2 + 2.2 = 17.4
3) Choose a scaling: 2 for the xaxis
and 5 for the yaxis will work fine.
4) Use the following window settings Xmin: 0
Xmax: 14.5
Xscl: 2
Ymin: 0
Ymax:17.4
Yscl: 2.5 We have two options for where to set the Ymin value.
If we use 13, we’ll get a big display of the graph, but lose the frame of reference with the origin.
If we use 0, we’ll keep our frame of reference (which can be very useful in applications),
but have a small display. Pre – Calculus Math 40S: Explained!
www.math40s.com 70 TRIGONOMETRY LESSON SEVEN
Part I Graphing Radian Functions Answers: 3. 2π
2π
300
=
= 2π ×
= 300
2π
b
2π
300
Since the sine pattern starts at 265 and the period is 300, we should extend our xaxis to 565.
1) Find the period: Period = 2) Find the minimum and maximum:
Minimum = d – a = 6.2 – 20.1 = 13.9
Maximum = d + a = 6.2 + 20.1 = 26.3 5) Graph the equation 3) Choose a scaling: 100 for the xaxis,
and 10 for the yaxis.
4) Use the following window settings
Xmin: 0
Xmax: 565
Xscl: 100
Ymin: 14
Ymax:27
Yscl: 10 4.
1) Find the period: Period = 2π
2π
=
= 11.11
b 0.18π 2) Find the minimum and maximum:
Minimum = d — a = 17 — 3.2 = 13.8
Maximum = d + a = 17 + 3.2 = 20.2 5) Graph the equation 3) Choose a scaling, 2 for the xaxis and
5 for the yaxis. 4) Use the following window settings
Xmin: 0
Xmax: 11
Xscl: 2
Ymin: 0
Ymax:27
Yscl: 5 Pre – Calculus Math 40S: Explained!
www.math40s.com 71 TRIGONOMETRY LESSON SEVEN
Part II Solving Radian Equations Example 1: The height of an object is given by the equation:
h(t) = 2.2sin0.123π (t +1.7)+15.2
Find the height after 2.4 seconds.
Alternatively, We may evaluate a point on a trig function by using the TI83.
h(2.4) = 2.2sin0.123π (2.4 +1.7)+15.2 (Radian Mode) h(2.4) =17.4 Questions: Evaluate the following functions for the value indicated. we could draw
the graph and
use:
2nd Trace
Value
x = 2.4 1) Evaluate h(t) =13.2cos 2π
(t 101)+6.5 when t = 105.
342 2) Evaluate h(t) = 20.1sin 2π
(t  265)+ 6.2 when t = 296
300 3) Evaluate h(t) =18.5cos 2π
(t  28)+ 4.5 when t = 45
365 Answers: π 4) Evaluate h(t) =12sin (t  3) 3 when t = 4
2 1) 19.7
2) 18.4
3) 22.2
4) 9 Pre – Calculus Math 40S: Explained!
www.math40s.com 72 TRIGONOMETRY LESSON SEVEN
Part II Solving Radian Equations Sometimes the yvariable is given, and we need to find the xvariable. In this case, we must
graph both the left side and right side of the equation, then find the intersection point.
Example 1: The height of an object is given by the equation: h(t) = 2.2sin0.123π (t +1.7) +15.2
Find the time when the object first reaches a height of 17. In your graphing calculator, graph:
y1 = 2.2sin0.123π (t +1.7)+15.2
y2 = 17
Now find the first intersection point.
TI83: 2nd Trace Intersect
The xvalue of the first intersection point
will be the time the object first reaches a
height of 17. Questions: Find the time when each of the following heights are reached.
Don’t forget to set your window properly!
1) Find the time when a height of 12 is first reached in the equation:
2π
h(t) = 13.2cos
(t 101) + 6.5
342 2) Find the time when a height of 23 is first reached in the equation:
2π
h(t) = 20.1sin
(t  265) + 6.2
300 3) Find the time when a height of 12 is first reached in the equation:
2π
h(t) = 18.5cos
(t  28) + 4.5
365 4) Find the time when a height of 0 is first reached in the equation:
h(t) = 12sin π
2 (t  3)  3 Pre – Calculus Math 40S: Explained!
www.math40s.com 73 TRIGONOMETRY LESSON SEVEN
Part II Solving Radian Equations Answers:
1) 2) t = 38.9 s 3) t = 12.3 s 4) t = 95 s t = 0.84 s Pre – Calculus Math 40S: Explained!
www.math40s.com 74 ...
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This note was uploaded on 02/09/2011 for the course 168 comm 168 taught by Professor Kiscabean during the Winter '10 term at UCLA.
 Winter '10
 KISCABEAN

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