7-4 Evaluating and
Sine and Cosine
and Cosines of Special Angles
45o and 60o angles are used many times in
urge you to memorize, or at least be able to derive
the sine and cosine of these special angles.
In a 30-60-90 triangle, the sides are in ratio of 1:
Look at the triangle below:
30o = y/r = 1/2, while the Cos 30o = x/r
Sin 60o = y/r = ,
while the Cos 60o = x/r = 1/2
In a 45-45-90 triangle, the sides are in ratio of 1
: 1 :
Study the triangle below:
45o = x/r = ,
while Cos 45o =
The wise person
memorize the following chart:
|| Sin q
|| Cos q
The graph of Sine
y = Sin x
of Sine Graph (Manipula Math)
Notice that this
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.
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
of Cosine Graph (Manipula Math)
This graph is
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,
and quad 4 back to positive. The difference in these two graphs
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
know their 5 important points. The zeros, maximum and minimum
The Sine curve has zeros at the
middle and end of a cycle. The maximum happens at the 1/4 mark
the minimum appears at the 3/4 mark.
The Cosine curve begins and ends with the
It has a minimum at the middle point. Zeros appear at the 1/4 and
3/4 mark of the cycle.
All angles can be
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
angle a, simply use the chart above to
the angle q.
q =120,then you are in quadrant II.
use the formula 180 - 120 to get a reference angle of 60.
q = 195, then you are in quadrant
Thus, use the formula 195 - 180 to get a reference angle of 15.
q = 300, then you are in quadrant IV.
Thus, use the formula 360 - 300 to get a reference angle of 60.
this idea of reference angles and Sine and Cosine is easy.
the reference angle as we did above and put the correct sign on each
From previous sections the Sine function
II and negative
in quadrants III
The Cosine function is positive
in quadrants I
II and III.
Sin ( 180o - 135o) = Sin 45o
Cos (360o - 310o) = Cos 50o
Sin (210o - 180o) = - Sin 30o (Sin is
negative in third quad)
Cos (180o - 112o) = - Cos 68o (Cos is
negative in 2nd quad)
Using your TI-82 to
Sine and Cosine
1) To calculate in
a) Press mode button and highlight degrees. Press enter.
b) Type in problem as example: Sin 45. Press enter.
c) Answer: .7071067812
Most trig answers are round to 4 decimal places. You can set your
calculator to fixed mode by: press mode. Press down arrow
highlight 4. Press enter. Press clear. Now
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 45o
Answer is .7071 if you are
still in fixed mode!!
Procedure 2 works regardless of the mode you are in.
2) To calculate in
a) Press mode button. Press down arrow twice.
radian. Press enter. You are now in rads. All
will now be calculated in rads.
b) Enter a problem. Example Cos 5. Press enter.
Answer (rounded to 4
places) is .2837
Procedure 2. (Using the rad button)
a) Enter a problem such as Sin -2
2nd function. Press angle button. Press down twice or 3.
b) Your display should look like: sin -2r
Answer is -.9093
Note: The use of the
radian button will override the setting in mode. Just like the
We should be ready to take a look at the
Take me back for some brush-up work!!