Objective: To graph functions that are the sums or differences of two sine functions, two cosine functions, or a sine and a cosine function.
"The function y = 2x + 4 is actually the sum of two functions, y = 2x and y = 4. For every value in the domain, there is a corresponding value in the range so that the point (x, 2x + 4) is on the graph of y = 2x + 4. The sums of trigonometric functions can also be graphed."
Resource: Hayden, Jerome D. and Hall, Bettye C. (1993). Trigonometry. Prentice Hall: Englewood Cliffs, New Jersey.
I. Graphing by addition of ordinates on Graphing Calculator.
1. Graph y = sinx over the interval [2p, 2p]. Then, on the same coordinate plane (but in a different color), graph y = cosx over the same interval. Finally, on the same coordinate plane (but in a different color), graph y = sinx + cosx over the same interval. For each value in the domain, how do the values of the range of y = sinx + cosx compare with the values of the ranges of y = sinx and y = cosx?
2. Graph y = 2cosx over the interval [2p, 2p]. Then, on the same coordinate plane (but in a different color), graph y = cos2x over the same interval. Finally, on the same coordinate plane (but in a different color), graph y = 2cosx + cos2x over the same interval. For each value in the domain, how do the values of the range of y = 2cosx + cos2x compare with the values of the ranges of y = 2cosx and y = cos2x?
II. Graphing by addition of ordinates by hand.
1. Open an Excel file. Label the first column "xcoordinates," the second column "y = 2sinx," the third column "y = sin0.5x," and the fourth column "y = 2sinx  sin0.5x." Set each column to round to four decimal places.
2. Set up the first column to range from 2p to 2p by intervals of p/4.
3. Fill in the second and third columns by utilizing the equations y = 2sinx and y =0.5sinx, respectively.
4. Fill in the fourth column by simply adding the second and third columns.
5. Graph y = 2sinx + sin0.5x by using the xcoordinates from column A and the ycoordinates from column D.
I.
1. y = sinx is in red, y = cosx is in blue, and y = sinx + cosx is in purple
Answer: At each xcoordinate, the ycoordinates of y = sinx + cosx is the sum of the ycoordinates of y = sinx and y = cosx.
2. y = 2cosx is in red, y = cos2x is in blue, and y = 2cosx + cos2x is in purple
Answer: At each xcoordinate, the ycoordinates of y = sinx + cosx is the sum of the ycoordinates of y = sinx and y = cosx.
II.
2. In cell A2, enter "=2*pi()". In cell A3, enter "=A2+pi()/4". Drag to fill down to row 18.
3. In cell B2, enter "=2*sin(A2)". Drag to fill down to row 18. In cell C2, enter "=1*sin(A2/2)". Drag to fill down to row 18.
4. In cell D2, enter "=B2+C2". Drag to fill down to row 18.
Here is a sample table:




6.2832  0  0  0 
5.4978  1.4142  0.3827  1.7969 
4.7124  2  0.7071  2.7071 
3.9270  1.4142  0.9239  2.3381 
3.1416  0  1  1 
2.3562  1.4142  0.9239  0.4903 
1.5708  2  0.7071  1.2929 
0.7854  1.4142  0.3827  1.0315 
0  0  0  0 
0.7854  1.4142  0.3827  1.0315 
1.5708  2  0.7071  1.2929 
2.3562  1.4142  0.9239  0.4903 
3.1416  0  1  1 
3.9270  1.4142  0.9239  2.3381 
4.7124  2  0.7071  2.7071 
5.4978  1.4142  0.3827  1.7969 
6.2832  0  0  0 
5. Here is a sample graph:
Graph each of the following functions (by hand!) over the interval [2p, 2p].
1. y = sinx + 2cosx
2. y = 3sinx + sin2x
3. y = 2sinx  cosx
4. y = cos2x + sin2x
1.
2.
3.
4.