74 Evaluating and
Graphing
Sine and Cosine
Sines
and Cosines of Special Angles
30^{o},
45^{o} and 60^{o} angles are used many times in
mathematics.
I strongly
urge you to memorize, or at least be able to derive
the sine and cosine of these special angles.
In a 306090 triangle, the sides are in ratio of 1:
:2
Look at the triangle below:
Sin
30^{o} = y/r = 1/2, while the Cos 30^{o} = x/r
=
Sin 60^{o} = y/r = ,
while the Cos 60^{o} = x/r = 1/2
In a 454590 triangle, the sides are in ratio of 1
: 1 :
Study the triangle below:
Sin
45^{o} = x/r = ,
while Cos 45^{o} =
The wise person
will
memorize the following chart:
Degrees 
radians 
Sin q 
Cos q 
0 
0 
0 
1 
30 
p/6 
1/2 

45 
p/4 


60 
p/3 

1/2 
90 
p/2 
1 
0 
The graph of Sine
and
Cosine Functions
y = Sin x
Demonstration
of Sine Graph (Manipula Math)
Notice that this
graph
is a periodic graph. It repeats the same graph every 2punits.
It is increasing from 0 to half pi, decreasing from half pi to negative
1.5 pi and increasing to 2 pi. Then the repeat starts. This
matches what happens to the Sine function in the quadrants.
Positive
in first and second and negative in the third and fourth. Maximum
value for the graph is 1 and the minimum value is 1.
y = Cos x
Demonstration
of Cosine Graph (Manipula Math)
This graph is
similar
to the previous shape. It is also a periodic graph with the cycle
being 2p. It also matches the signs of
the quadrants with quad one being positive, quads two and three,
negative
and quad 4 back to positive. The difference in these two graphs
is
the starting point for the Cosine graph. It starts at the maximum
value. The Sine curve started at the origin point.
An easy way to remember these graphs is
to
know their 5 important points. The zeros, maximum and minimum
points.
The Sine curve has zeros at the
beginning,
middle and end of a cycle. The maximum happens at the 1/4 mark
and
the minimum appears at the 3/4 mark.
The Cosine curve begins and ends with the
maximum.
It has a minimum at the middle point. Zeros appear at the 1/4 and
3/4 mark of the cycle.
Reference
Angles
All angles can be
referenced
back to an angle in the first quadrant. This is true because the
trig functions are periodic. Study each of the quadrant formulas
below to find the reference angles.
To find the
reference
angle a, simply use the chart above to
locate
the angle q.
Example:
If
q =120,then you are in quadrant II.
Thus,
use the formula 180  120 to get a reference angle of 60.
Example:
If
q = 195, then you are in quadrant
III.
Thus, use the formula 195  180 to get a reference angle of 15.
Example:
If
q = 300, then you are in quadrant IV.
Thus, use the formula 360  300 to get a reference angle of 60.
Relating
this idea of reference angles and Sine and Cosine is easy.
Determine
the reference angle as we did above and put the correct sign on each
function.
From previous sections the Sine function
is positive
in
quadrants I
and
II and negative
in quadrants III
and IV.
The Cosine function is positive
in quadrants I
and IV,
while
negative in
quadrants
II and III.
Examples
Sin 135^{o}
=
Sin ( 180^{o}  135^{o}) = Sin 45^{o}
Cos 310^{o}
=
Cos (360^{o}  310^{o}) = Cos 50^{o}
Sin 210^{o}
=
Sin (210^{o}  180^{o}) =  Sin 30^{o} (Sin is
negative in third quad)
Cos 112^{o}
=
Cos (180^{o}  112^{o}) =  Cos 68^{o} (Cos is
negative in 2nd quad)
Using your TI82 to
evaluate
Sine and Cosine
1) To calculate in
degrees:
Procedure 1
a) Press mode button and highlight degrees. Press enter.
b) Type in problem as example: Sin 45. Press enter.
c) Answer: .7071067812
Note:
Most trig answers are round to 4 decimal places. You can set your
calculator to fixed mode by: press mode. Press down arrow
and
highlight 4. Press enter. Press clear. Now
enter
the problem: Sin 45
Answer is now .7071
Procedure 2. (Use the degree button)
a) Enter problem: Sin 45
b)Press 2nd function. Press "angle" button. You get a menu
with the degree symbol as choice 1. Either press 1 or enter.
c) Your display should now look like this: Sin 45^{o}
Press enter.
Answer is .7071 if you are
still in fixed mode!!
Note:
Procedure 2 works regardless of the mode you are in.
2) To calculate in
radian
measure.
Procedure 1
a) Press mode button. Press down arrow twice.
Highlight
radian. Press enter. You are now in rads. All
problems
will now be calculated in rads.
b) Enter a problem. Example Cos 5. Press enter.
Answer (rounded to 4
decimal
places) is .2837
Procedure 2. (Using the rad button)
a) Enter a problem such as Sin 2
Press
2nd function. Press angle button. Press down twice or 3.
b) Your display should look like: sin 2^{r}
Press enter.
Answer is .9093
Note: The use of the
radian button will override the setting in mode. Just like the
degree
button does.
We should be ready to take a look at the
other
trig functions!!
Take me back for some brushup work!!