Math Exam #1

use a calculator to approximate the value of a trigonometric function: 6.2

**mode: select radians or degrees; add degrees: 2nd and angle (apps)** a) cos(43°) ≈ 0.73135: just plug into calculator (insert degrees sign) b) tan(5) ≈ -3.38 (put into radians if degrees aren't written) c) sec(103°) ≈ -4.45 (reciprocal of cos, calculator: 1/cos(103°)) d) cot(square root 2/2) ≈ 1.17 (reciprocal of tangent, calculator: 1/tan(square root 2/2))

find the exact values of the trigonometric functions for integer multiples of pi/4, pi/6, and pi/3: 6.2

*look at example #9 and #10 find the exact value of 3csc(2pi/3) + cot(3pi/4) 3csc(2pi/3) + cot(3pi/4) = 3 • 2/square root 3 + (-1) = 2square root 3 - 1

use even-odd properties to find the exact values of the trigonometric functions: 6.3

*look at picture at top of pg. #7 f(-ø) = f(ø) is EVEN f(-ø) = -f(ø) is ODD sin(-ø) = -sinø = odd csc(-ø) = -cscø = odd cos(-ø) = cos ø = even sec(-ø) = sec ø = even tan(-ø) = -tanø = odd cot(-ø) = -cotø = odd

determine the domain and range of the trigonometric functions: 6.3

- sineø and cosineø = x & y: domain: all real numbers; range: [-1,1]; - secantø = 1/cosø = *problem whenever cosø=0: domain: {ø | ø ≠ pi/2 + kpi, k integer}; range: (-infinity, -1] U [1, infinity) - cosecant = 1/sinø: domain: {ø | ø ≠ kpi, k integer}; range: (-infinity, -1] U [1, infinity) - tangentø = sinø/cosø: domain: {ø | ø ≠ pi/2, kpi integer}; range: (-infinity, infinity) (all real numbers) - cotangentø = cosø/sinø: domain: {ø | ø ≠ kpi, k integer}; range: (-infinity, infinity) (all real numbers)

graphing thing to keep in mind

-w causes a reflection (flip pattern)

key conversion factors for linear speed & angular speed: 6.1

1 radian/radius distance; radius distance/1 radian; 2pi radians/1 revolution; 1 revolution/2pi radians

how many revolutions is 2pi radians: 6.1

1 revolution = 360°

how many revolutions is pi radians: 6.1

1/2 revolution = 180°

how many revolutions is pi/2 radians: 6.1

1/4 revolution = 90°

secø: 6.2

1/x

cscø: 6.2

1/y

cosø: 6.2

= x

sinø: 6.2

= y

finding area of a circular sector: 6.1

A=1/2r2(squared)ø (A=area; r=radius (needs to be squared); ø=theta=angle (needs to be in radians))

graph functions of the form y = Acos(w(x+B)) + C using transformations: sine & cosine

For cosine graph: A=amplitude (how high or low the arcs of the graph can go); cos=type of graph; w=what you divide 2pi (the period) by; B=phase shift (affects x-axis); C=vertical shift (up or down)(affects y-axis); want to find: amp, reflection, & center for x-axis; period, space btwn key pts, phase shift, & tick marks for y-axis & type of graph (sine, cosine, etc. for whole graph) amp: A number cos: starts high (top amplitude point) goes to the right middle, down, middle, up: this is one period for cosine (big U) starting point can change depending on shifts and etc. reflection: negative sign in front of A, changes pattern to start down (bottom amplitude point), middle, up, middle, down center: y=top amplitude point from 0, unless the C(vertical shift) then starting point for y-axis: changes it (-2 is 2 down and this is now center, and 2 is 2 up and this is now center, then base amplitude off of these new center points) period: 2pi/w spacing btwn key pts: period/4; phase shift: center starting point for x-axis: is the letter B in equation: to write: multiply w to x and B (distribute or undistribute depending on what looking for) tick marks: need the common denominator between the key point spacing and the phase shift. Use this denominator to determine the tick marks for the x-axis of graph, tick marks on y-axis are determined by amplitude and center point

graph functions of the form y = Asin(w(x+B)) + C using transformations: sine & cosine

For sine graph: A=amplitude (how high or low the arcs of the graph can go); sin=type of graph; w=what you divide 2pi (the period) by; B=phase shift (affects x-axis); C=vertical shift (up or down)(affects y-axis); want to find: amp, reflection, & center for x-axis; period, space btwn key pts, phase shift, & tick marks for y-axis & type of graph (sine, cosine, etc. for whole graph) amp: A number sin: starts at the center origin (0,0); goes to the right up, middle, down, middle: this is one period for sine: starting point can change depending on shifts and etc. reflection: negative sign in front of A, changes pattern to down, middle, up middle center: y=0, unless the C(vertical shift) then starting point for y-axis: changes it (-2 is 2 down and 2 is 2 up and these are now center, then base amplitude off of these new center points) period: 2pi/w spacing btwn key pts: period/4; phase shift: center starting point for x-axis: is the letter B in equation: to write: multiply w to x and B (distribute or undistribute depending on what looking for) tick marks: need the common denominator between the key point spacing and the phase shift. Use this denominator to determine the tick marks for the x-axis of graph, tick marks on y-axis are determined by amplitude and center point

determine the signs of the signs of the trigonometric functions in a given quadrant: 6.3

QI: x>0, y>0: all trig functions +, (1,1); QII: x<0, y>0: sine and cosine +, (-1,1); QIII: x<0, y<0: tangent and cotangent +, (-1,-1); QIV: x>0, y<0: cosine and secant +, (1,-1)

domain and range: secant & cosecant

SECANT domain and range: d: {x | x ≠ one specific VA (vertical asymptote) location + k • period/2} r: (-infinity, # bottom facing graphs highest point is at] U [# top facing graphs lowest point is at, infinity) COSECANT domain and range: d: {x | x ≠ one specific VA (vertical asymptote) location + k • period/2} r: (-infinity, # bottom facing graphs highest point is at] U [# top facing graphs lowest point is at, infinity) - same

domain and range: sine & cosine

SINE domain and range: d: all real numbers (-infinity, infinity) r: [lowest negative point, highest positive point] COSINE domain and range: d: all real numbers (-infinity, infinity) r: [lowest negative point, highest positive point] - same

domain and range: tangent & cotangent

TANGENT domain and range: d: {x | x ≠ pi/2 + k•pi} r: all real numbers (-infinity, infinity) COTANGENT domain and range: d: {x | x ≠ 0 + k • pi} r: all real numbers (-infinity, infinity) - not the same

find the exact values of the trigonometric functions of quadrantal angles: 6.2

a) cos(7pi/2) = 0 * look at picture on pg. 3 b) tan540° = 0 * look at picture on pg. 3

s = ø; t = ø: 6.2

angle corresponds to arc; ø = 0: =0-90°: (1,0); ø = pi/2: =90-180°: (0,1); ø = pi: =180°-270°: (-1,0); ø = 3pi/2: =270°-0°: (0, -1)

negative angle: 6.1

angle made by clockwise rotation (——> to the right, below initial side)

positive angle: 6.1

angle made by counter-clockwise rotation (<—— to the left, above initial side)

find the exact values of the trigonometric functions using a point on the unit circle: 6.2

ex) P = (4/5, -3/5); sint=y= -3/5; cost=x= 4/5; tant=y/x= -3/5/4/5 = -3/4; csct=1/y= -5/3; sect=1/x= 5/4; cott=x/y= 4/5/-3/5 = -4/3

determine the amplitude and period of sinusoidal functions: sine & cosine

example #7: write the equation of a sine function that has amplitude 4 and period 3, phase shift: pi/4 start with this form: 4sin(2pi/3(x+pi/4)) to find full amplitude: 2pi/w = period # given, then solve for w: in this case, you get 2pi/3 then, change form: 4sin(2pi/3(x+pi/4)) to 4sin(2pi/3x + pi2(squared)/6) by distributing w to x and B

find an equation for a sinusoidal graph: sine & cosine

example #8: find several equations for each graph find center in between top and bottom amplitude points and plug in for C count distance from center and top or center and bottom and that is the amplitude, plug in for A look at the pattern of graph to determine if it's sine or cosine to find period: 2pi/w = period you can see (look at pattern and see how it lays) for example #8 it is 2pi/w = 2pi, w = 1, plug this into equation to find phase shift: if center is still on center y-axis, no phase shift, if center shifted right or left on x-axis, then that is phase shift, to the right(positive number) is minus #, and to the left(negative number) is plus #, plug into B look for reflections: if pattern is flipped then there is a reflection and if pattern is normal then there is no reflection end up with something like: y=-3sin(x+pi)+2

determine the period of trigonometric functions: 6.3

f(ø+p) = f(ø) - sinø = f(ø) = period: 2pi - cosø = f(ø) = period: 2pi - cscø = f(ø) = period: 2pi - secø = f(ø) = period: 2pi - tanø = f(ø) = period: pi - cotø = f(ø) = period: pi

standard position: 6.1

initial (starting) side on the positive x-axis

graph sinusoidal functions using key points: sine & cosine

look at previous graphing instructions but use sinusoidal functions to find amp, period, phase shift and vertical shifts from basic equation form and go from there using basic instructions

example #2: 6.2

look in notes

converting from radians to degrees: 6.1

multiply by 180°/pi rad

Converting from degrees to radians: 6.1

multiply by pi rad/180°

properties of cotangent function: tangent & cotangent

odd/even? odd symmetry: origin period: pi

properties of tangent function: tangent & cotangent

odd/even? odd symmetry: origin period: pi

find the exact values of the trigonometric functions of pi/4: 6.2

pi/4 = 45° *look at picture on pg. 3 P = (square root 2/2, square root 2/2) sin(pi/4) = square root 2/2 cos(pi/4) = square root 2/2 tan(pi/4) = 1 csc(pi/4) = square root 2 sec(pi/4) = square root 2 cot(pi/4) = 1

find the exact values of the trigonometric functions of pi/6 and pi/3: 6.2

pi/6 = 30° *look at picture on pg. 4 P = (square root 3/2, 1/2) sin(pi/6) = 1/2 cos(pi/6) = square root 3/2 tan(pi/6) = square root 3/3 csc(pi/6) = 2 sec(pi/6) = 2 square root 3/3 cot(pi/6) = square root 3 pi/3 = 60° *look at picture on pg. 5 P = (1/2, square root 3/2) sin(pi/3): square root 3/2 cos(pi/3): 1/2 tan(pi/3): square root 3 csc(pi/3): 2 square root 3/3 sec(pi/3): 2 cot(pi/3): square root 3/3

angular speed: 6.1

radians/unit time; used to calculate the distance the body covers in terms of rotations or revolutions to the time taken. Speed is all about how slow or fast an object moves. Angular speed is the speed of the object in rotational motion.

find the values of the trigonometric functions using fundamental identities: 6.3

reciprocal identities: 1. cscø = 1/sinø, 2. secø = 1/cosø, 3. cotø = 1/tanø quotient identities: 1. tanø = sinø/cosø, 2. cotø = cosø/sinø pythagorean identities: 1. cos2(squared)ø + sin2(squared)ø = 1, 2. 1 + tan2(squared)ø = sec2(squared)ø 3. cot2(squared)ø + 1 = csc2(squared)ø other written down: 1. 1 = sec2(squared)ø - tan2(squared)ø

Finding Arc Length of a Circle: 6.1

s=rø (requires angle in radians)(s=arc length; r=radius; ø=theta,=inner angle in radians)

graph functions of the form y = Asec(w(x+B)) + C and y = Acsc(w(x+B)) + C: secant & cosecant

secant(sec): y = sec x = 1/cos x; graph cosine graph first then add vertical asymptotes and graph based on "humps" of graph(up hump means graph up, down hump means graph down in between asymptotes) - asymptotes are where cosine graph hits middle line (center line) - normal graph goes to the right and down then up cosecant(csc): y = csc x = 1/sin x; graph sine graph first then add vertical asymptotes and graph based on "humps" of graph - asymptotes are where sine graph hits middle line - normal graph goes to the right and up then down

find the exact values of trigonometric functions of an angle given one of the functions and the quadrant of the angle: 6.3

sinø: y/r cosø: x/r tanø: y/x cscø: r/y secø: r/x cotø: x/y ex) if sinø = - 5/13 and ø is in quadrant III, find the values of the remaining trigonometric functions of ø: x<0 & y<0; x = -12, y = -5, r = 13 sinø: -5/13, cosø: -12/13, tanø: 5/12, cscø: -13/5, secø: 13/-12, cotø: 12/5

Graph functions of the form y = Atan(w(x+B)) + C and y = Acot(w(x+B)) + C: tangent & cotangent

tangent(tan): y = tan(x) *period: pi* - still need to find center point, "amplitude", period (w), and phase shift - use sine and cosine ideas for basic graph concepts - tangent pattern: low, middle, high - the tangent graph "starts" with a center point at the phase shift location - no phase shift = "start" at (0,0) cotangent(cot): y = cotx *period: pi* - still need to find center point, "amplitude", period (w), and phase shift - use sine and cosine ideas for basic graph concepts - cotangent pattern: high, middle, low - the cotangent graph "starts" with a V.A. at the phase shift location - no phase shift = "start" with V.A. on y-axis

linear speed: 6.1

unit distance/unit time; the measure of the concrete distance travelled by a moving object. The speed with which an object moves in the linear path is termed as linear speed. In easy words, it is the distance covered for a linear path in the given time

cotø: 6.2

x/y

to find x or y for trigonometric functions: 6.2

x2(squared) + y2(squared) = 1

tanø: 6.2

y/x