29 Notes - M 408N –Calculus for Scientists Dr. Olena...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: M 408N –Calculus for Scientists Dr. Olena Bormashenko Notes 8/29 We will use radians instead of degrees 2 pi rad = 360 deg pi rad = 180 deg angle (a) = 180/pi * theta theta = a * pi /180 Radians for 60 deg & 90 deg: theta = 60 * pi /180 = pi/3 rad theta = 90 * pi/180 = pi/2 rad Degrees for 2pi/3 radians: a=180/pi * theta = 180/pi = 2pi/3 = [pi cancel so (180 * 2)/3 = 120] = 120 degrees Visualizing trig functions: Draw a line theta radians away from the position x axis (standard position) cos theta = x coordinate sin theta = y coordinate < theta = (cos theta, sin theta) Use the unit circle (google image) to determine cos pi/2, cos pi, and sin 0 cos pi/2 = 0 ; (x coordinate of 90 degree angle) cos pi = -1 (x coordinate of 180 degree angle) sin 0 = 0 ( y coordinate of 0 degree angle) tan theta = sin theta / cos theta (y/x) cot theta = cos theta / sin theta (x/y) csc theta = 1/sin theta (1/y) sec theta=1/cos theta (1/x) For theta between 0 and pi/2 we have the following expressions sin theta = opposite/hypotenuse ; opposite from the angle** cos theta = adjacent/hypotenuse ; adjacent to the angle** tan theta = opposite/ adjacent cot theta = adjacent/ opposite csc theta = hypotenuse/ opposite sec theta = hypotenuse / adjacent Find all the trig functions of the angle pi/4 ( 45 degrees) sin theta = 1/root 2 cos theta = 1/root 2 tan theta = 1 cot theta = 1 csc theta = root 2 sec theta = root 2 Trig identities: sin^2 theta + cos^2 theta =1 tan^2 theta + 1 = sec ^2 theta 1+ cot^2 theta = csc ^2 theta Identities [sin & cos] sin (-theta) = -sin theta --> odd function cos (-theta) = cos theta --> even function Adding and Subtracting : sin (x+y) = sin x cos y + cos x sin y cos (x+y) = cos x cos y - sin x sin y tan (x+y) = tan x + tan y / (1-tan x tan y) sin (x-y) = sin x cos y - cos x sin y *rest of the addition and subtraction identities are found in the book (Appendix D; A29) Find all the trig functions of theta given tan theta = 3/4pi and pi < theta < 2 pi 3rd quadrant because tangent is positive there (All in 1st, sine is + in 2nd, tan is + in 3rd, and cos is + in 4th quad) tan theta = sin theta/cos theta = 3/4 opposite = 3 adjacent = 4 hypotenuse = 5 3^2 + 4^2 = 5^2 sin theta = -3/5 cos theta = -4/5 csc theta = -5/3 sec theta = -5/4 cot theta = 4/3 Exponential functions: f(x)= a^x is called an exponential function a^x = a* a * a * a …… * a..[x times] a^0=1 if x = 0 if x is negative then a^-x = 1/a^x IF x is a rational number x = p/q where p & q are integers and q > 0 a^x= a^(p/q) = q root a^p Laws of Exponents: 1. a^x a^y = a^(x+y) 2. a^x/a^y= a^(x-y) 3. (a^x)^y= a^xy 4. (aB)^x = a^x B^x 2^3 + 2^5 = 2^ 8 2*2*2 * 2*2*2*2*2 = 2^8 Graphs from book pgs 54-56 e = 2.718281828…… ...
View Full Document

This note was uploaded on 11/21/2011 for the course M 408N taught by Professor Gualdini during the Spring '10 term at University of Texas.

Ask a homework question - tutors are online