Evaluating and Graphing Sine and Cosine

7-4 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 30-60-90 triangle, the sides are in ratio of 1: √3 :2


Look at the triangle below:



Sin 30^o = y/r = 1/2, while the Cos 30^o = x/r = √3/ 2

Sin 60^o = y/r = √3/ 2, while the Cos 60^o = x/r = 1/2


In a 45-45-90 triangle, the sides are in ratio of 1: 1 : √2


Study the triangle below:



Sin 45^o = x/r = √2/ 2, while Cos 45^o = √2/ 2


The wise person will memorize the following chart:

Degrees	Radians	Sin q	Cos q
0	0	0	1
30	π/6	1/2
45	π/4
60	π/3		1/2
90	π/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 2 pi units. 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 2 pi. 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 TI-82 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.