Trig test 2
The value of tan^-1X is an angle in the interval
(-pi/2,pi/2)
What is the range of y=tan^−1x?
(-pi/2,pi/2)
In the function y=tan^−1x, the x in tan^−1x is defined over the interval
(−∞,∞)
Sec t=
1/x
Csc t=
1/y
If the terminal side of an angle lies in Quadrant I, then
All of the trigonometric functions are positive.
Reference angle case 1
Case 1: If the terminal side of θC lies in quadrant I, then theta θR=θC.
Reference angle case 2
Case 2: If the terminal side of θC lies in quadrant II, then θR=π−θC.
Reference angle case 3
Case 3: If the terminal side of θC lies in quadrant III, then θR=θC−π.
Reference angle case 4
Case 4: If the terminal side of θC lies in quadrant IV, then θR=2π−θC
Properties of graph y=cotx
D: R: (−∞,∞) Period: pi Odd y intercept: None Zeros of the form ((2N+1)/2)pi, where n is an integer Every halfway point has a y-coordinate of: −1 or 1 Principle cycle: (0,pi)
Properties of graph y=cosx
D: (-infinity,infinity) All real numbers R: [-1,1] Period: 2pi Even y intercept: 1 X intercept: (2n+1)*pi/2, where n is an integer relative max: X=2piN, where N is an integer relative min: X=Pi+ 2piN, where N is an integer Zeros of the form ((2N+1)/2)pi, where n is an integer
Properties of graph y=sinx
D: (-infinity,infinity) All real numbers R: [-1,1] Period: 2pi Odd y intercept: 0 X intercept: npi, where n is an integer relative max: x= pi/2 + 2piN, where N is an integer relative min: 3pi/2 + 2piN, where N is an integer Zeros of the form: Nπ, where n is any integer
Properties of graph y=tanx
D: (-infinity,infinity) All real numbers, except odd integer multiples of pi/2 R: (−∞,∞) Period: pi Odd y intercept: 0 X intercept: Zeros of the form Nπ, where n is any integer Every halfway point has a y-coordinate of: −1 or 1 Principle cycle: (-pi/2,pi/2)
Properties of graph y=secx
D: all real numbers except odd integer multiples of π/2 R: (−∞,−1]∪[1,∞) Period: 2pi Even relative max: x=πn where n is an odd integer Vertical asymptotes of the form: x= nπ/2 where n is an odd integer
Properties of graph y=cscx
D: all real numbers except odd integer multiples of π/2,pi, R: (−∞,−1]∪[1,∞) Period: 2pi Odd relative max: x= -pi/2 + 2piN, where N is an integer Vertical asymptotes of the form: x=nπ where n is an integer
Which of the following statements is not true about the function y=Acos(Bx)?
If B>0, then the function y=Acos(−Bx) is equivalent to the function y=−Acos(Bx).
Graph of y=cos x
Quarter points: (0,1), (pi/2,0), (pi, -1), (3pi/2,0), (2pi, 1)
Which of the following statements describes the definition of amplitude of a sine or cosine function?
The amplitude is the measure of half the distance between the maximum and minimum values.
If theta= sine^-1x, then which of the following statements best describes angle θ?
The angle θ is an angle satisfying the inequality -π/ 2 ≤ θ ≤ π/2 having a terminal side lying in Quadrant I, Quadrant IV, on the positive x-axis, on the positive y-axis, or on the negative y-axis.
If θ=cos^−1x, then which of the following statements best describes angle θ?
The angle θ is an angle satisfying the inequality 0≤θ≤π having a terminal side lying in Quadrant I, Quadrant II, on the positive x-axis, on the positive y-axis, or on the negative x-axis
If θ=tan^−1x, then which of the following statements best describes angle θ?
The angle θ is an angle satisfying the inequality −π/2 < θ < π/2 having a terminal side lying in Quadrant I, Quadrant IV, or on the positive x-axis.
Given the expression Csc (cos^-1(-1/2)) which of the following is not true?
The expression Csc (cos^-1(-1/2)) is equal to -2/square root of 3
If A, B, and C are constants such that B>1, then which of the following statements is true about the graph of y=Acos(Bx−C)?
The period is 2π/B and the phase shift is C/B.
If A, B, C, and D are constants, then which of the following statements is true about the graph y=Acos(Bx−C)+D?
The range is [−IAI+D, IAI+D].
Which of the following is not a characteristic of the sine function?
The sine function obtains a relative maximum at x equals x= π/2+πn where n is an integer.
When sketching the graph of y=Atan(Bx+C)+D which of the following best describes how to determine the x-coordinates of the halfway points of the principal cycle?
The x-coordinate of each halfway point is located halfway between the x-coordinate of the center point and a vertical asymptote.
Which of the following statements is true?
The x-intercepts of y=tanx are the same as the x-coordinates of the center points of y=tanx.
Which of the following is not a characteristic of the cosine function?
The y-intercept is 0.
The graph y=sin(x+C) can be obtained by horizontally shifting each quarter point of y=sin(x) to the left C units.
Which of the following statement best describes the graph of y=sin(x+C) where Upper C greater than 0C>0?
In the function y=cos^−1x, the x in cos^−1x is defined over the interval
[-1,1]
In the function y=sin^−1x, the x in sin^−1x is defined over the interval
[-1,1]
The value of sine^-1X is an angle in the interval
[-pi/2,pi/2]
The value of cos^-1X is an angle in the interval
[0,pi]
What is the domain of the restricted cosine function whose inverse function is y=cos^−1x?
[0,pi]
If 3π/4<θ<5π/6, then which of the following mathematical statements is true?
cos^−1(cosθ)=θ
Graph of y=sin x
quarter points: (0,0),(pi/2,1),(pi,0),(3pi/2,-1),(2pi,0)
Secant/Sec
r/x= Hyp/Adj
Cosecant/Csc
r/y= Hyp/Opp
Which of the following expressions does not result in an angle having a terminal side that lies in Quadrant I?
sin^-1(sin6pi/5)
Which of the following expressions is equivalent to the angle pi/6?
tan^-1(tan(-5pi/6))
tan: the angle lies in the interval 0<theta<pi/2
terminal side: Quad 1
sin: the angle lies in the interval 0<theta<pi/2
terminal side: Quad 1 or positive y-axis
tan: the angle lies in the interval -pi/2<theta<0
terminal side: Quad 4
sin: the angle lies in the interval -pi/2<theta<0
terminal side: Quad 4 or negative y-axis
cos: the angle lies in the interval pi/2<theta<pi
terminal side: Quadrant 2 or negative x-axis
cos: the angle lies in the interval 0<theta<pi/2
terminal side: Quadrant I or positive x-axis
If the terminal side of an angle lies in Quadrant IV, then
the Cosine (and secant) functions are positive
If the terminal side of an angle lies in Quadrant II, then
the Sine (and cosecant) functions are positive
If the terminal side of an angle lies in Quadrant III, then
the Tangent (and cotangent) functions are positive.
Value of B
to graph divide period into 4 and then add that number to each x value.
Value of A
to graph multiple Y value of quarter points by A.
Cos t=
x/1
Cosine/Cos
x/r= Adj/Hyp
Cot t=
x/y
Cotangent/Cot
x/y= Adj/Opp
Sin t=
y/1
Sine/Sin
y/r= Opp/Hyp
Tan t=
y/x
Tangent/Tan
y/x= Opp/Adj
y=-2cos(-4x) is the same as
y=-2cos(4x)
y=sin(-pix) is the same as
y=-sin(pix)
Phase shift:
y=Asin(Bx+C) C/B
Amplitude
y=AsinX IAI absolute value of A so always positive. Changes range
Period
y=sin(BX) Period= 2pi/B as long as B is positive
Finding the principle cycle of y=Atan(Bx−C)+D,
−π/2<Bx−C<π/2
