Trig
If a=4, then the coordinates of the focus are
(3,6)
If a>0, the equation of the parabola is of the form
(x-h)^2=4a(y-k)
The graph of y = sec x is symmetric with respect to the _________ and has vertical asymptotes at ______________
y-axis, odd multiples of π/2
B. False. If a=4, then the equation of the directrix is
y=-2
For the hyperbola y^2/16-x^2/81=1, the asymptotes are
y=4x/9, y=-4x/9 (the asymptotes are given by y-k=plus or minus a/b(x-h)
It is easiest to graph y = sec x by first sketching the graph of _______.
y=cos x
sin^2(θ/2) = (1-cosθ)/2 HALF ANGLE FORMULA
θ/2
The equation of the hyperbola is of the form _______.
(y-k)^2/a^2-(x-h)^2/b^2=1
For a hyperbola, the foci lie on a line called the
transverse axis
True or false. For y = 3sin(πx) the amplitude is 3 and the period is π/2.
False. The amplitude is 3 and the period is 2. (2π/π = 2)
True or false. The equation sine θ = 2 has a real solution that can be found using a calculator.
False. The range of the sine function is [-1,1]. Since 2 is not from the range of the sine function, the equation sine θ = 2 does not have a real solution.
True or false. Exact values can be found for the sine of any angle.
False. The results from plane geometry can be used to find the exact value of the sine function of the selected acute angles not all.
Two acute angles whose sum is a right angle are called __________
complementary angles.
The function f(θ)=secθ is not defined for ______________ multiples of ____________
odd integer, π/2 (90°)
The function f(θ)=tanθ is not defined for ______________ multiples of ____________
odd integer, π/2 (90°)
y = sin^-1x means __________, where −1≤x≤1 and −π/2≤y≤π/2
x=siny
Area of a circle of radius r is
πr^2
What is the range of the tangent function?
(-∞,∞)
The unit circle has center ________ and radius ___. The equation of the unit circle is __________
(0,0) , 1, x^2+y^2=1
If the center of the hyperbola is (2,1) and a=3, then the coordinates of the vertices are
(2,-2)(2,4)
The function y = A sin (ωx), A>0, has amplitude 3 and period 2; then A=_____ and ω=____
3, π
In a hyperbola, if a=3 and c=5, then b =
4
Exact values of the six trigonometric functions of π/2 can be obtained using which special triangle?
45°-45°-90°
For the ellipse x^2/25 +y^2/9=1, the value of a is 5, the value of b is 3, and the major axis is the x-axis.
Clear
If the center of an ellipse is (3,−4), the major axis is parallel to the x-axis, and the distance from the center of the ellipse to its vertices is a=5 units, then the coordinates of the vertices are (8,-4) and (-2,-4)
If an ellipse is parallel to the x-axis, it has center (h,k) and vertices (h+a,k) and (h-a,k). Therefore, the given ellipse, with center (3,-4) has vertices with coordinates (3+5,-4)=(8,4)and(3-5,4)=(-2,-4).
The distance d between two points P1=(1x 1,y 1) and P2=(x2,y2) =
StartRoot (x2-x 1)^2+(y2-y1)^2 EndRoot .
True or false. sin(-θ)+sinθ=0 for any value of θ
True.
degrees, minutes, and seconds (DMS)
The unit of measure for describing latitude and longitude. A degree is 1/360th of a circle. A degree is further divided into 60 minutes, and a minute is divided into 60 seconds. A minute is also 1/3600 of a degree.
True or false. y= tan^-1 x means x=tany, where -∞<x<∞ and −π/2<y<π/2
True.
Domain of y=sinx
The set of all real numbers.
True or false. Two solutions of the equation sinθ = 1/2 are π/6 and 5π/6
The statement is true because sineπ/6 = 1/2 and sin 5π/6 = 1/2
True or false. The set of all solutions of the equation tanθ=1 is given by StartSet θ | θ = π/4+kπ, k is any integer.
The statement is true because the tangent function has the period π.
In the polar coordinates (r,θ) r can be negative.
True
True or false. The angular speed ω of an object traveling around a circle of radius r is the angle theta (measured in radians) swept out, divided by the elapsed time t.
True
True or false. Exact values can be found for the trigonometric functions of 60°
True. Using the Pythagorean theorem and the fact that each angle of the equilateral triangle measures 60°, the exact values for the trigonometric functions of 60° can be found.
Cotangent
adjacent/opposite (reciprocal of tangent)
When you look up at an object, the acute angle measured from the horizontal to a line-of-sight observation of the object is called the ________.
angle of elevation
Pythagorean Theorem
a²+b²=c²
Which of the following functions is even?
cosine | Note that a function f is even if f(-θ) = f(θ) for all θ in the domain of f. According to the even-odd properties of trigonometric functions, cos (-θ) = cos θ, same with sec.
Two angles in standard position that have the same terminal side are ______________
coterminal
One period of the graph of y = sin(Ωx) or y = sin(Ωx) is called a ___________
cycle
If a person is looking down at an object, the acute angle measured from the horizontal to a line of sight to the object is called the angle of ________
depression
A function for which f(x)=f(−x) for all values of x in the domain of f is called a(n) _________ function.
even
Secant
hypotenuse/adjacent
If sin θ = 0.2, then sin (-θ) = _____ and sin (θ+2π) = ______
-0.2, 0.2
True or false. In establishing an identity, it is often easiest to just multiply both sides by a well-chosen nonzero expression involving the variable.
False
Decide whether the following equation is true or false. tan π/10 = cot (9π/10)
False, because tanθ = cot (π/2-θ). That means tan π/10 = cot(π/2 -π/10) = cot(2π/5).
True or false. It is impossible to obtain exact values for the inverse secant function.
False.
True or false. cos (π/2-θ)=cosθ
False.
True or false. sin (α+β)=sinα+sinβ+2sinαsinβ
False.
True or false. tan(2θ)+tan(4θ)=tan(4θ)
False.
True or false. sin(2θ) has two equivalent forms: 2sinθcosθ and sin^2θ-cos^2θ.
False. Only 2sinθcosθ for double angle formula sin.
A ______ _______ is a positive angle whose vertex is at the center of a circle.
central angle
Which function is positive in quadrant IV?
cos θ
Which of the following equations does not equal 1?
cos^2θ-sin^2θ
To find the inverse secant of a real number x such that |x| ≥ 1, convert the inverse secant to an inverse _______.
cosine
Cosecant
hypotenuse/opposite
Formula for circumference of a circle
C = 2πr
180°=
π radians
True or false. The only trigonometric functions with a domain of all real numbers are the sine and cosine functions.
True, because only for sine and cosine, there is no concern about dividing by 0, so θ can be any angle.
For the parabola y^2 = 4ax, the line segment joining the two points (a,2a) and (a,-2a) is called the ________________
latus rectum
The origin in rectangular coordinates coincides with the ________ in polar coordinates; the positive x-axis in rectangular coordinates coincides with the __________________ in polar coordinates.
pole, polar axis
If the radius of a circle is r and the length of the arc subtended by a central angle is also r, then the measure of the angle is 1 _______.
radian
If θ is an acute angle and secθ=44 then cosθ =1/4θ.
True, because secθ = 1/cosθ.
For the point with polar coordinates (1,-π/2 ), which of the following best describes the location of the point in a rectangular coordinate system?
on the y-axis
The graph of y = tan x is symmetric with respect to the _________ and has vertical asymptotes at __________
origin, odd multiples of π/2
The reference angle is always an acute angle.
True, because according to the definition of a reference angle, if θ denotes an angle that lies in a quadrant then the acute angle formed by the terminal side of θ and the x-axis is called the reference angle for θ
Which of the following intervals describes where sine^-1(sin x) = x?
−π/2<x<π/2
Exact values of the six trigonometric functions of π/3 can be obtained using which special triangle?
30°-60°-90°
True or false. sin(sin^-1 (0)) = 0 and cos(cos^-1 (0)) = 0
True
True or false. tan(2θ)=2tanθ/1-tan^2θ
True
True or false. tanθ*cosθ=sinθ for any θ not equal to (2k+a)pi/2
True
Decide whether the following equation is true or false.
True, because sin/cos = opposite/adjacent and tan = opposite/adjacent, thus tan = sin/cos.
True or false. The graph of y=cosx is decreasing on the interval [0,π].
True.
True or false. sin(-θ)+cos(-θ)=cosθ-sinθ
True.
True or false. The graphs of y = sin x and y = cos x are identical except for a horizontal shift.
True. The sine and cosine functions are sinusoidal functions. That means when the graph of y = cos x is shifted to the right π/2 units to obtain the graph of y = cos(x-π/2), the graph is same as the graph of y = sin x.
True or false. The graphs of y = tan x, y = cot x, y = sec x, y = csc x each have infinitely many vertical asymptotes.
True. Vertical asymptotes of y = tan x and y = sec x occur at all odd multiples of π/2, and vertical asymptotes of y = cot x and y = csc x occur at all integer multiples of π. .
Let t be a real number and let p=(a,b) be the point on the unit circle that corresponds to t. Then sine t = _____ and cosine t = ____
b, a
For any angle θ in standard position, let p=(a,b) be any point on the terminal side of θ that is also on the circle x^2 +y^2 = r^. Then sin θ = _____ and cos θ = _____
b/r, a/r
The sine and __________ functions are cofunctions.
cosine (The functions sine and cosine, tangent and cotangent, and secant and cosecant are called cofunctions of each other.)
A(n) ____________ is the collection of all points in the plane such that the distance from each point to a fixed point equals its distance to a fixed line. The line through the focus and perpendicular to the directrix is called the _______________
parabola, axis of symmetry
For an angle θ that lies in quadrant III, the trigonometric functions _________ and _________ are positive.
tangent, cotangent
The sum formula for tangent function is tan (α+β) =
tanα+tanβ/(1-tanαtanβ)
If P is a point with polar coordinates (r,θ), the rectangular coordinates (x,y) of P are given by x=_______and y=_______.
x=rcosθ and y = rsinθ
To graph y=3 sin(−2x) using key points, the equivalent form _________________ could be graphed instead.
y =−3sin(2x)
If sin α = ±√(1-cosθ)/2, then which of the following describes how the value of θ is related to the value of α.
θ = 2α
Suppose θ=π/2 is the only solution of a trigonometric equation in the interval 0≤θ<2π. Assuming a period of 2π, which of the following formulas gives all solutions of the equations, where k is any integer?
θ=π/2 + 2kπ
tan^2θ-sec^2θ=
-1
Range of y=sinx
-1 ≤ y ≤ 1
cos(-θ)-cosθ=
0
The maximum value of y =sine x, x≤0≤2π, is _______ and occurs at x =________
1, π/2
The sine, cosine, cosecant, and secant functions have period _____; the tangent and cotangent functions have period _____. .
2π, π
The function y=−3cos(6x) has amplitude ____________ and period _____________.
3, π/3
Which of the following is not in the range of the sine function?
3/2. The range of the sine function consists of all real numbers between -1 and 1, inclusive.
What are the domain and the range of y=secx?
Domain: The set of all real numbers, except odd integer multiples of π/2. Range: The set of all real numbers less than or equal to -1 or greater than or equal to 1.
For circular motion on a circle of radius r, linear speed equals angular speed divided by r.
False
SOHCAHTOA
sine = opposite over hypotenuse, cosine = adjacent over hypotenuse, tangent = opposite over adjacent
y=sec^-1 means
According to the definition of inverse secant function, y=sec^-1 means x=secy, where domain is -1≤x≤1 (|x| ≥ 1) and the range is 0 ≤ y ≤ π, y ≠ π/2
True or false. sin^2θ=1-cos^2θ
True.
The polar coordinates of a point are unique.
False
Suppose that f and g are two functions with the same domain. If f(x)= g(x) for every x in the domain, the equation is called a(n) _______. Otherwise, it is called a(n) _______ equation.
identity, conditional
tan(θ/2)=(1-cosθ)/(sinθ) HALF ANGLE FORMULA
sinθ
Most trigonometric equations have unique solutions.
False.
Determine whether the statement below is true or false. csc^-1(0.6) is not defined.
True. The inverse cosecant function y = csc^-1 is defined for |x| ≥ 1.
If a particle has a speed of r feet per second and travels a distance d (in feet) in time t (in seconds), then d =
rt
On a circle of radius r, a central angle of θ radians subtends an arc of length s=_____; the area of the sector formed by this angle θ is A =_______.
rθ, 1/2r^2θ
An object travels around a circle of radius r with constant speed. If s is the distance traveled in time t around the circle and θ is the central angle (in radians) swept out in time t, then the linear speed of the object is v = _____ and the angular speed of the object is ω = ______.
s/t, θ/t
For a triangle with sides a, b, c, and opposite angles A, B, C, the Law of Sines states that
sinA/a=sinB/b=sinC/c
cos(2θ)=cos^2θ- __________ = ___________ -1 = 1 - _______________
sin^2θ=2cos^2θ-1=1-2sin^2θ
An angle θ is in ________ _______ if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive x-axis.
standard position
