Assignment 85

 "We don't know a millionth of one percent about anything." - (Thomas Alva Edison) The Maya lived long ago in southern Mexico and Guatemala. Ancient records suggest they were active and productive over 5,000 years ago. While we use a base 10 number system, the Maya people used base 20. Our number 43, for instance, means 4(10) + 5(1). Using our digits in the Mayan system, the number 43 would be written as 23, meaning 2(20) + 3(1). The Maya devised an accurate calendar that had 18 months of 20 days each, plus 5 additional days, to make 365 days. If you think about it, our number system creates numbers using the symbols (digits) 0,1,2,3,4,5,6,7,8,9 If you had a base 20 system, you would need 20 symbols, since our numbers 10,11,12,13,14,15,16,17,18,19 would need to be represented by a symbol. While the Mayans didn't use these symbols, a base 20 system could be 0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f,g,h,i,j In this system, our number 312, which is (15)(20) + 12, could be written ec What did Herkimer do when he decided to get off his feet and play a game of Scrabble? Answer: He sat down for a spell. Herky wants to know: Is intergull calculus the high-level mathematics discussed by seabirds? Is the shortest distance between two jokes a straight line? ASSIGNMENT #85 Reading: Section 13.2, pages 776-779. Written: Page 783/(Quiz 1) 1-18.
 Mathematical discoveries:HERON'S FORMULA: Given any triangle with sides a, b, and c, let s = (a+b+c)/2. The Egyptian mathematician Heron of Alexandria (ca 75 A.D.) established that the area of the triangle is SQRT[s(s-a)(s-b)(s-c)]. Consider a right triangle with sides 3,4,5. The area would be (1/2)(3)(4) = 6 (sq. units). Using Heron's Formula, s = (3+4+5)/2 = 6 and the area would be SQRT[6(6-3)(6-4)(6-5)] = SQRT[36] = 6 (sq. units). Note that Heron's formula works for any triangle, not just right triangles.
Key thoughts:

In the proper context, one can speak of positive and negative angles. That is, -30o has meaning in the proper context.

You can measure a length in feet or inches. And, there is a relationship between the units. We know 1 ft. = 12 inches. (Note that the units are important. People will look at you in a funny way if you write 1 = 12.) In the same vein, angles can be measured in degrees or radians. The relation is 360o = 2p radians. Note that 2p is approximately 2(3.14) = 6.28, so there are about 6.28 radians in 360o. 1 radian is approximately 57.3 degrees.

Radian measure is very useful in higher levels of mathematics. It is much easier to work with radians than degrees when you work with the amazing branch of mathematics called calculus. Even when working with circles, radians are useful. For instance, the arc length formula

s = rq

is simple, but it assumes that the central angle q is measured in radians. If the angle q is measured in degrees, the formula becomes

s = 2pr(q/360) = p rq/180

which is less easy to work with.

Review thoughts....

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.

 Problem: If q is an angle such that sin q = -7/25 and cos q = 24/25, find the values of the remaining for trig functions. Solution (with communication): Tan q = sin q /cos q = (-7/25)/(24/25) = -7/24. Cot q = 1/tan q = -24/7. Csc q = 1/sin q = -25/7. Sec q = 1/cos q = 24/25. Problem: A circle has radius 8 inches. What is the length of an arc of the circle intercepted by a central angle of 1.4 radians? What is the area of the sector determined the angle? Solution (with communication): Since the central angle is measured in radians, the requested arc length is (8)(1.4) = 11.2 (in.). The area of the sector is (1/2)(82)(1.4) = 44.8 (sq. in.).