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# Trigonometry, Week 3 MTH 123 08

GVSU

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This 5 page Class Notes was uploaded by eshumaker1328 on Thursday September 15, 2016. The Class Notes belongs to MTH 123 08 at Grand Valley State University taught by Dr Meyering in Fall 2016. Since its upload, it has received 4 views. For similar materials see Trigonometry in Mathematics at Grand Valley State University.

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Date Created: 09/15/16

Monday, September 12, 2016 Trigonometry 1.5: Four Additional Circular Functions: Tanget, Secant, Cosecant, and Cotangent Functions - Tangent: tan X = sin X/cos X • ^ produces: tangent functions : {(x, tan x)} - Reciprocal Functions • SECANT: sec x = 1/cos x — produces: secant function : {(x, sec x)} • COSECANT: csc x = 1/sin x — produces: cosecant function : {(x, csc x)} • COTANGENT: cot x = cos x/ sin x — produces: cotangent function : {(x, cot x)} - E = is an element of… [can be] - Z = intergers (+/-) • K E Z FUNCTIONS DOMAIN RANGE {(x, tan x)} x doesn’t = pi/2 +k(pi) all real numbers {(x, sec x)} x doesn’t = pi/2 +k(pi) | sec x | >/= 1 {(x, csc x)} x doesn't = k(pi) | csc x | >/= 1 {(x, cot x)} x doesn't = k(pi) all real numbers x cos x sin x tan x sec x csc x cot x 0 1 0 0 1 undeﬁned undeﬁned pi/6 sqrt 3/2 1/2 1/sqrt3 2/sqrt3 2 sqrt3 1 1 pi/4 1/sqrt2 1/sqrt2 sqrt2 sqrt2 pi/3 1/2 sqrt3/2 sqrt3 2 2/sqrt3 1/sqrt3 pi/2 0 1 undeﬁned undeﬁned 1 0 pi -1 0 0 -1 undeﬁned undeﬁned 1 Monday, September 12, 2016 x cos x sin x tan x sec x csc x cot x 3pi/2 0 -1 undeﬁned undeﬁned -1 0 1.6: Negative Identities and Periods for the Circular Functions - Negative Identities [x E all real numbers in the domain of the function] • cos(-x)= cos x • sin(-x)= -sin x • tan(-x)= -tan x • sec(-x)= sec x • csc(-x)= -csc x • cot(-x)= -cot x - Period Function [f(x)] • f = period function - simple harmonic motion • a point on a pendulum that is swinging back and forth, or the bobbing up and down of a weight attached to a spring, repeats the movement in equal intervals of time • is periodic 2.1: Graphs of the Sine and Cosine Functions - Sine Function • wave: the connection of the points with a smooth continuous curve to produce the graph of the sine function a more precise way of saying ^ is a sinusoidal wave • repeats every 2pi • over one period is called a cycle • 2 Monday, September 12, 2016 • the endpoints are the critical points • the x-intercepts referred to as zeros or roots • y=sin x is called amplitude - amplitude = .5 |max y-coord. - min y-coord| x 0 pi/6 pi/4 pi/3 pi/2 3pi/4 pi 5pi/4 3pi/2 7pi/4 2pi sin x 0 1/2 1/sqrt2 sqrt3/2 1 1/sqrt2 0 -1/ -1 -1/ 0 sqrt2 sqrt2 - Cosine Function x 0 pi/6 pi/4 pi/3 pi/2 3pi/4 pi 5pi/4 3pi/2 7pi/4 2pi cos x 1 sqrt3/2 1/sqrt2 1/2 0 -1/ -1 -1/ 0 1/sqrt2 1 sqrt2 sqrt2 - Important Characteristics of Graphs (y=sin x & y= cos x) • common - sinusoidal waves with period 2pi • amplitude = .5 |1-(-1)| = 1 • each cycle has 4 identically shaped arc sections - intervals along x-axis are 1/4 • endpoints of the four arcs are critical points • different y = sin x y = cos x for 0 </= x </= 2pi, ﬁrst arc is increasing, the for 0 </= x </= 2pi, ﬁrst two arcs are decreasing next two arcs are decreasing, and the last arc and the next two arcs are increasing is increasing. y int = (0,0) y int = (0,1) x int: x = kpi x int: x = (pi/2) + kph symmetric with respect to the origin symmetric with respect to the y-axis - call each graph pure form 3 Monday, September 12, 2016 - vertical stretch: multiplying the funct. by a real number greater than 1 or less than -1 - vertical shrink: multiplying the funct. by a nonzero real number between -1 & 1 Amplitude of Sinusoidal Wave in the Form y = A cos x and y = A sin x - |A| = amplitude - |A| > 1 produces a vertical stretch - |A| < 1 produces vertical shrink - A < 0 produces a reﬂection across the x-axis Graphing Calculator/ CAS - Mode: Radian - Window: [example] • Xmin: 0 Xmax: 4pi Xscl: pi/2 • Ymin: -6 Ymax: 6 - Then input your function Characteristics - continuous periodic waves - four identically shaped arcs for each cycle with critical points at each end - the x-intercepts and symmetry - amplitude or reﬂection changes to the maximum and minimum functional values SINUSOIDAL FUNCTIONS Function Domain Range X-intercepts Amp. Symmetry Period y = sin x x E all real numbers |sin x| </= 1 x = kpi A = 1 origin P = 2pi y = cos x x E all real numbers |cos x| </= 1 x = (pi/2) + kpi A = 1 y-axis P = 2pi 4 Monday, September 12, 2016 f(x) = x^2 -f(x) = -x^2 — ﬂips over x axis f(-x) =(-x^2)— ﬂips over y axis cf(x) =c x^2— like Asin[B(x - C)] + D f(cx) = Asin[B(x - C)] + D B — how much period/cycle occurs in 2π [IS NOT THE PERIOD] 2π/B = new period A — amplitude C — horizontal shifts D — vertical shifts Period is the max to the new max f(x + c) — horizontal move left f(x - c) — horizontal move right f(x) + c —vertical move up f(x) - c —vertical move down 5

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