  Assignment 87

 "Geometry will draw the soul toward the truth and create the spirit of philosophy" - (Plato, ca 400 B.C.) "Let no one ignorant of geometry enter this door."(Motto on entrance to Plato's Academy) Gottfried Leibniz (Germany, 1646-1716): This man is, today, accepted as a coinventor of the amazing branch of mathematics we know as the calculus. Yet, ironically, he was, in the words of a contemporary, "buried like a robber." His funeral was attended by just one person, his secretary. Several years passed before his gravestone was even inscribed. A conflict with Isaac Newton (1642-1727) over who discovered (created) calculus ended poorly for Leibniz when he appealed to England's Royal Society to resolve the dispute. The President of the Royal Society happened to be (you guessed it) Isaac Newton. A report prepared by the Newton-appointed commission officially accused Leibniz of plagiarism, effectively ostracizing him from his academic circles before he died in 1716. Why did Herkimer take a hammer to the barn loft when he was sleepy? Answer: So he could hit the hay. Herky's friends: PHIL M. UP: A gas station attendant. LEE FLETT: A publisher of small pamphlets. JUDGE KNOTT: A man of the law who refused to criticize others. ASSIGNMENT #87 Reading: Review Section 13.3, as necessary. Written: Page 789/61-68. In each case, sketch the angle and the unit circle. Find the x and y coordinates of the intersection of the terminal side of the angle and the circle. Relate these coordinates to the trigonometric functions of the angle. Mathematical fact:MAY 28, 585 B.C. : Thales, one of the seven wise men of antiquity, supposedly predicted an eclipse of the sun on that day. Modern technology has allowed us to establish that an eclipse of the sun actually did happen on that day. This is the earliest historical event known to the exact day.
An angle q is in standard position if

(a) its vertex is at the origin, and
(b) one side (the initial side) coincides with the positive x-axis.
(c) the other side (the terminal side) simply needs to be a ray with (0,0) as its starting point.

The KEY to trigonometry:

All values of the sine and cosine functions can be found on the unit circle.

Example:

Sketch the angle 315o in standard position. Draw the unit circle with center at (0,0). The point at which the terminal side of the angle intersect the unit circle is (1/÷2, -1/÷2). The sine and cosine of the angle 315o are the y and x coordinates, respectively. That is

sin 315o = -1/÷2, and

cos 315o = 1/÷2

We can also conclude that tan 315o = -1, csc 315o = -÷2 ,sec 315o = ÷2 , and cot 315o = -1.

 Problem: If q is an angle measured in radians such that 0 < q < 2 p and if tan q is undefined, what are the possible values of q ? Solution (with communication): On the unit circle with q in standard position, tan q = y/x. Hence tan q is undefined whenever x = 0. That is, tan q is undefined when the terminal side of the angle q in standard position lies along the y-axis. Subject to the restrictions 0 < q < 2 p , the possible values of are p /2 radians and 3 p /2 radians. Problem: If q is an angle in standard position whose terminal side contains the point (3,-4), find the values of the six trigonometric function of q. Solution (with communication): The length of the segment joining (0,0) to (3,-4) is 5. Hence sin q = -4/5, cos q = 3/5, tan q = -4/3, csc q = -5/4, sec q = 5/3, and cot q = -3/4.