Applied Knowledge Assessment
Reduction Formula
Assignment
Part 1: Experiment
Click here to see an interactive simulator that allows you to graph for various a and b values.
In the lower window on the left side of the screen, you can see the graphs of two functions: (yellow curve) and (blue curve). The values of the amplitudes of the curves, a and b, can be set to any number between -2 and 2.
The value of can also be changed using the interactive ruler on the bottom of the screen. For this assignment, set =1, so that you will be working with the functions and .
The top window on the left side of the screen shows the graphs of two different functions. The green curve is the graph of the sum. The red curve is the graph of cosine function. A is the amplitude of the cosine wave and is the phase shift.
Note: You can change the shape and location of the red curve by using the two scales next to the top window to change the values of the amplitude and the phase shift. Observe how the curve responds to the changes in A and.
You may realize that it is possible to set A and to values that cause the red curve to coincide precisely to the green curve. This means that there are values of A and that make. Your goal is to figure out what these values are.
To figure out the values, use the window to the right of the screen. Click the square under window marked (a, b). The point on the coordinate plane corresponding to the ordered pair (a, b) will appear.
Click the square marked (A,). The red line segment that appears has the length A, and it makes an angle of radians with the horizontal axis.
If you change the values of A and again, you will see the red line segment change in length and rotate.
The table below lists various values for a and b. Set a and b for the values listed on the table. For each case, adjust A and until the red curve coincides with the green curve. Read the resulting values for A and from the scale and record them in the table below. Also, note the position of the red line segment when the green and red curves coincide. Record your observations in the table as well.
(a, b) / A / / tan / Record your notes about the position of the red line segment here.(1, 1) / 1.41 / 45 / 1 / The position of the red light segment is creating a right triangle with the blue ling segment with the angles being 90,45 ,and 45.
(1, -2) / 2.25 / -63 / -1.96 / The position of the red light segment is creating a right triangle with the blue ling segment with the angles being 90,63 ,and 27.
(-1, 0.5) / 1.12 / 153 / -0.51 / The position of the red light segment is creating a right triangle with the blue ling segment with the angles being 90,73 ,and 17.
Reflection: From you observations, make a conclusion about how to find A and using a and b. Show how your conclusion works for the values of A and you obtained using the simulator.
Say, for example, we were given the tangent of an angle and A. A can be thought of as the length of the line between the origin and the point (a,b). Since we also have the angle through arctan(tan(@)) it is easy to calculate the rest.
Arctan(1)=45
sin(45)*1.41=a=1
cos(45)*1.41=b=1
Part 2: Algebra
You can derive the reduction formula algebraically using the result you obtained in Part 1 as a clue.
The goal is to rewrite as a cosine function with amplitude A. Start by multiplying and dividing a and b in by A expressed in terms of a and b:
=
Factor out A:
=
Write the coefficients in front of cost and sint as two trigonometric functions of angle.The coefficient in front of costshould be cosine and the coefficient in front of sintshould be sine:
Using the cosine of the difference formula from Unit 2 Lesson 10, rewrite the resulting expression as a single cosine:
=
Record the expressions for A, ,,and in terms of a and b.