Trig final
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)
Sec T=
1/x
Csc T=
1/y
Isosceles
A triangle that has 2 equal sides.
Equilateral
A triangle with three congruent sides
Area of a sector formula
A= 1/2 * r^2 * theta Ex: 1/2 * 90 feet^2 * 1/3= 1350 ft^2
Finding central angle
A=1/2 * r^2 * theta EX. A= 70ft; R= 10ft 2 (70)= 2 [1/2(10^2) theta] 140= 100 theta theta= 7/5 radians
Pythagorean Theorem
A^2+B^2= C^2 OR X^2+Y^2= R^2
If the terminal side of an angle lies in Quadrant I, then
All of the trigonometric functions are positive.
Which of the following statements is not true concerning radian measure?
An angle in standard position having a radian measure of theta = -11pi/6 has a terminal side that lies in Quadrant IV.
Which of the following statements best describes an angle that is in standard position?
An angle is in standard position if the vertex is at the origin of a rectangular coordinate system and the initial side lies along the positive x-axis.
For any real number t, if P(x,y) is a point on the unit circle corresponding to t, then which of the following does not accurately define a trigonometric function?
Cot t= y/x, x =/ 0
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 best describes the term general angles?
General angles are angles that are not restricted in size and can be either positive angles, negative angles, or zero.
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).
Which of the following statements is not true? Choose the correct answer below
If the central angle of a sector of a circle is θ=5° and the radius of the circle is r=2 cm, then the arc length of the sector of the circle is 10 cm. NOT 5 DEGREES BUT 5 X 180/PI
Which of the following statements is true?
If θ is an angle belonging to π/6, π/3, π/4 families, then the reference angle will be π/6, π/3, π/4, respectively.
Graph of y=cos x
Quarter points: (0,1), (pi/2,0), (pi, -1), (3pi/2,0), (2pi, 1)
A 10 inch (in diameter) pizza is cut into various sizes. What is the area of a piece that was cut with a central angle of 32°?
REMEMBER TO DIVIDE THE DIAMETER INTO 2 TO GET THE RADIUS
Length of arc
S= R*Theta
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.
Which of the following statements is not true concerning angle measure?
The angle in standard position formed by rotating the terminal side of angle one complete counterclockwise rotation has a radian measure of π radians.
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.
Which of the following statements is not true? Choose the correct answer below.
The area of a sector of a circle is given by the equation A= 1/2 * θ * r^2, where θ is an angle given in degrees. NOT DEGREES, RADIANS
Which of the following statements is true?
The length of the leg opposite either of the π/4 angles of a special π/4, π/4, π/2 right triangle with a hypotenuse of SquareRoot 2 is equal to 1.
Which of the following statements is true about a special right triangle?
The length of the leg opposite the π/3 angle of a special π/6, π/3, π/2 right triangle is equal to the square root of 3 times the length of the leg opposite the π/6 angle
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.
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.
Which is true about the unit circle?
The unit circle is symmetric about the x-axis, y-axis, and the origin.
Which of the following statements is not true?
There are infinitely many points that lie on the graph of the unit circle that have integer coordinates.
Which of the following angles does not belong to the π/4 family of angles?
Theta= 10pi/4
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.
Which of the following statements is true concerning the conversion between degree and radian measure?
To convert from radians to degrees, multiply by 180 degrees and divide by π.
Which of the following statements best describes two coterminal angles?
Two angles in standard position are coterminal if they have the same terminal side
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?
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]
Scalene
a triangle with no congruent sides
Cofunction identities (Cos)
cos (pi/2- theta) = sin theta
Which of the following is not a fundamental identity?
cos theta= tan theta/ sin theta
Cofunction identities (Cot)
cot (pi/2 - theta) = tan theta
Trigonometric expression (Cot)
cot theta= cos theta/ sin theta
Suppose that a right triangle has an acute angle θ and side lengths of hyp, opp, and adj. Which of the following does not accurately define a trigonometric function?
cot theta= opp/adj
Trigonometric expression (Cot and csc)
cot^2theta+1=csc^2theta
Cofunction identities (Csc)
csc (pi/2 - theta) = sec theta
Which of the following is not a valid equation?
csc pi/6 = cos pi/3
Trigonometric expression (Csc)
csc theta= 1/sin theta
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
Cofunction identities (Sec)
sec (pi/2 - theta) = csc theta
Trigonometric expression (Sec)
sec theta= 1/cos theta
Which of the following is not a cofunction identity?
sec theta= cos (pi/2 - theta)
Which of the following is NOT a fundamental identity?
sec^2theta +1= tan^2theta
Cofunction identities (Sin)
sin (pi/2-theta) = cos theta
Trigonometric expression (Sin and cos)
sin^2theta+cos^2theta=1
Cofunction identities (Tan)
tan (pi/2 - theta) = cot theta
If the terminal side of an angle θ lies in Quadrant III, then which of the following is true?
tan theta > 0, sec theta < 0
Trigonometric expression (Tan)
tan theta= sin theta/cos theta
Trigonometric expression (Tan and sec)
tan^2theta+1=sec^2theta
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
Cos t=
x/1
Cosine/Cos
x/r= Adj/Hyp
Cot T=
x/y
Cotangent/Cot
x/y= Adj/Opp
The standard form of the equation of the unit circle is
x^2+y^2=1
Sin T=
y
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
