Sine Behavior Chart
We begin by remembering our sine behavior chart from last time.
Cosine Behavior Chart
We now continue our work by developing a behavior chart for the cosine function. Because the cosine is defined as the leg adjacent divided by the hypotenuse, we now will be keeping a close eye on the horizontal leg of each of the multiple right triangles as the angles rotate from 0 to 360. We see that at zero degrees, the right triangle will be compressed vertically, and the horizontal leg will be full length or 1. So, the cosine of zero degrees is 1. As the angle approaches 90 degrees, the triangle will be compressed horizontally and the cosine approaches zero.We can also see that in the first quadrant these horizontal legs will be decreasing from 1 to 0 at 90 degrees.
As we work on the second and third quadrants, we will have horizontal leg length that are left of the y-axis, and therefore will be considered to be negative. So as we move from 90 to 180 degrees we can "see" the horizontal leg going from length zero to length -1, therefore cosine is degreasing again (it was in the first also) in the second quadrant.
In quadrant three, it moves from -1 back to zero and is therefore increasing. In quadrant four, we go from 0 to 1 and are therefore still increasing.
Tangent Behavior Chart
Now we need a behavior chart for the tangent function. Again, we fall back on our geometry roots and remember the meaning of tangent. You remember that a tangent line was a line which intersected a circle in only 1 point, called the point of tangency. In our unit circle think of a line tangent to the unit circle at the point (1,0). If the hypotenuse is extended beyond the right triangle until it intersects this tangent line, the vertical length cut off by this extended hypotenuse using the point (1,0) as its endpoint is the tangent. (See below)
At zero degrees this tangent length will be zero. Hence, tan(0)=0. As our first quadrant angle increases, the tangent will increase very rapidly. As we get closer to 90 degrees, this length will get incredibly large. At 90 degrees we must say that the tangent is undefined (und), because when you divide the leg opposite by the leg adjacent you cannot divide by zero.
As we move past 90 degrees into the second quadrant, we will have to "back" up the hypotenuse extended so that it will meet with the orginal tangent line. Because the length now formed is below the x-axis, we can see that the tangent in the second quadrant is negative. This same negative result will happen again in the fourth quadrant. The tangent result in the second quadrant will quickly decrease in length. Because this decrease is in negatives, the tangent is again increasing. (as it always is)
In the third quadrant the hypotenuse extended will now meet the tangent line above the x-axis and is now positive again. At 270 degrees we again have an undefined (und) result because we cannot divide by zero..
Trigonometry students need to be able to produce these behavior charts in the minds so that they can complete the chart below with no calculator.