Assignment 84

"Who has not been amazed to learn that the function y = ex, like a phoenix rising again from its own ashes, is its own derivative." - (Francois le Lionnais)

Math History Tidbit:

Infinity: Throughout the history of mathematics, the concept of infinity seemed to reside in a "twilight zone." Amazingly, the mystery of the infinite wasn't really resolved until the time and work of Georg Cantor (1845-1918). Some of the great mathematicians of the past treated infinity as a number. For instance, Leonhard Euler (1707-1783) in his Algebra (1770), incorrectly stated that 1/0 equals infinity and then, without any clarification, went on to say that 2/0 is twice as large as 1/0. The Hindus, for all their magnificent contributions to our present number system, also demonstrated some confusion over the concept of infinity. Bhaskara (1114-1185) did suggest that if A is not zero, then A/0 is infinite, but went on to assert that (A/0)x0 = A

"Infinity is where things happen that don't."

-Anonymous student.

Herkimer's Corner

Why did Herkimer feed garlic to his dog?

Answer: He wanted the dog's bark to be worse than its bite.

Herky's friends:

SUE WIDGE: She worked for the Department of Sanitation.

AUNTIE DOTE: She was good at producing remedies to counteract the effects of poison.

ASSIGNMENT #84

Reading: Review Section 13.1, as necessary.

Written: Pages 772-774/15-17, 46-51.

Items for reflection:

Mathematical word analysis:
SINE: This word has a somewhat detailed and confusing history. The Hindu mathematician Aryabhata (ca. 500) referenced what we know as the sine function as ardhajya, which translates to "half-chord." Consider a unit circle. Draw a chord of the circle, then draw radii to the end points of the chord. Now, bisect the central angle formed, and let half of the measure of the central angle be x. Remember now... the radius of the circle is 1 unit. So sin (x) is actually the length of half of the chord. (Construct a diagram and convince yourself this is true.)

There are six trigonometric functions: sine, cosine, tangent, cosine, cosecant, secant, cotangent.

All values of y = sin x and y = cos x are found on the unit circle.

Sine and cosine are the most important of the functions in the sense that the other four trig functions can be defined in terms of these two:

tan A = sin A/cos A if cos A is not zero.

cot A = 1/tan A = cos A/sin A if sin A is not zero.

csc A = 1/sin A if sin A is not zero.

sec A = 1/cos A if cos A is not zero.

You should note that your calculator has SIN, COS, and TAN keys. Try experimenting with them. Use the MODE option to put your calculator in Degree mode. Then, if you calculate SIN(30), you should get 0.5. Also, COS(30) = 0.8660254038 and TAN(30) = 0.5773502692. Using right triangle trigonometry from geometry, you should remember that SIN(30) = 1/2, COS(30) = 3/2, and TAN(30) = 1/3. If you convert these ratios to decimals, you will see they agree with the respective calculator computations.

Problem: The legs of a right triangle are 8 in. and 15 in. (a) Find the length of the hypotenuse; (b) If x is the measure of the smallest angle of the triangle, find sin(x), cos(x), tan(x), csc(x), sec(x), and cot(x).

Solution (with communication):

(a) The length of the hypotenuse is SQRT[82 + 152] = SQRT[289] = 17 (inches).

(b) Sin(x) = 8/15 ==> csc(x) = 15/8.

Cos(x) = 15/17 ==> sec(x) = 17/15.

Tan(x) = 8/15 ==> cot(x) = 15/8.