On the Unit Circle, Where , When Is Undefined?

Sine Behavior Graph

We begin by remembering our sine behavior chart from last fourth dimension.

Cosine Behavior Chart

We forthwith continue our act by developing a behaviour chart for the cosine function. Because the cos is defined As the leg contiguous divided by the hypotenuse, we now will be guardianship a close-set eyeball 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 decent triangle will follow compressed vertically, and the horizontal leg will be full length operating theatre 1. So, the cos of zero degrees is 1. A the tilt approaches 90 degrees, the triangle will be flat horizontally and the cos approaches no.We can besides date that in the for the first time quarter-circle these horizontal legs will be allargando from 1 to 0 at 90 degrees.

As we work on the ordinal and third quadrants, we will have horizontal leg length that are left of the y-axis, and therefore bequeath be considered to be negative. And then as we move from 90 to 180 degrees we can buoy "see" the horizontal peg going from length naught to length -1, therefore cosine is degreasing again (it was in the original also) in the second right angle.

In quadrant three, IT moves from -1 back to nada and is thence increasing. In quadrant four, we go from 0 to 1 and are therefore still increasing.

Tangent Behavior Graph

Now we need a conduct chart for the tangent purpose. Again, we fall back on our geometry roots and remember the meaning of tangent . You remember that a tangent line was a argumentation which intersected a circuit in only 1 point, called the point of tangency. In our unit circle think up a line tangent to the unit circle at the breaker point (1,0). If the hypotenuse is sprawly beyond the right triangle until IT intersects this tangent line, the vertical length cut off by this extended hypotenuse victimization the level (1,0) as its endpoint is the tangent. (See below)

At zero degrees this tangent length will constitute zero. Hence, tan(0)=0. As our first quadrant angle increases, the tangent will step-up very rapidly. As we stick closer to 90 degrees, this length leave get incredibly large. At 90 degrees we must suppose that the tangent is indefinite (und), because when you divide the pegleg opposite by the leg adjacent you cannot divide by 0.

As we move past times 90 degrees into the second quadrant, we leave have to "plunk for" sprouted the hypotenuse extended thus that it will come across with the orginal tan line. Because the length now formed is below the x-axis, we can discove that the tangent in the second quadrant is negative. This Lapplander dissident result will happen again in the fourth quadrant. The tangent result in the second right angle bequeath quickly decrease in duration. Because this decrease is in negatives, the tan is once again increasing. (equally it always is)

In the third right angle the hypotenuse spread-eagle 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-bodied to produce these deportment charts in the minds so that they can out-and-out the chart below with none calculator.

Trigonometry Outline

Some Notes from Class 

The Tidy You Indigence for Calculus @ WLC

On the Unit Circle, Where , When Is Undefined?

Source: http://faculty.wlc.edu/buelow/PRC/ntT-4.htm

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