precalc Q's
what transformation of f is represented by g(x)= 3f(x)?
VD of 3
what transformation of function f is represented by g(x) = 3 f(x)?
VD of 3
describe the transformation: g(x)= 5f(x)
VD of 5
refer to fig. 3-2l on pg. 92; what is the vertical translation (sinusoidal axis)?
VT of 13
describe the transformation: g(x)= 4 + f(x)
VT of 4
what transformation of f is represented by h(x)= 5 + f(x)?
VT of 5
refer to triangle ABC: A = 38°, b = 4, c = 7; find the length of the third side
a = 4.5685... units
for right triangle ABC, if B is the right angle, then sin A = __?__
a/b
for right triangle ABC, if B is the right angle, side a^2 = ___?___ in terms of sides b and c
a^2 = b^2 - c^2
name these greek letters: α, β, γ, Φ
alpha, beta, gamma, phi
when you multiply two sinusoids with much different periods, you get a function with a varying ___?___
amplitude
why is y= 3x^5 not an exponential function even though it has an exponent?
an exponential function has a constant with a variable exponent
give a number x for which |x| = -x
any negative number
give a number x for which |x| = x
any nonnegative number; or 0
for triangle ABC, write the law of cosines involving angle B
b^2 = a^2 + c^2 - 2(a)(c) cos B
x^2 + y^2 = 9 is the equation of a(n)....
circle
cos (3x + 5x) = ___?___ in terms of functions of 3x and 5x
cos 3x cos 5x - sin 3x sin 5x
cos 7 cos 3 + sin 7 sin 3 = cos ___?___
cos 4
in terms of cosines and sines of 53° and 42°, cos (53° - 42°) = ___?___
cos 53° cos42° + sin 53° sin 42°
by the composite argument properties, cos (x - y) = ___?___
cos x cos y + sin x sin y
in general, cos (x + y) = ___?___ in terms of cosines and sines of x and y
cos x cos y - sin x sin y
in the composite argument properties, cos (x + y) = ___?___
cos x cos y - sin x sin y
sin (90°- θ) = cos (___?___)
cos θ
if sin θ = 5/13 and angle θ is in Quadrant II, what is cos θ?
cos θ = -12/13
state the pythagorean property for cosine and sine
cos^2x + sin^2x = 1
what is the pythagorean property for cosine and sine?
cos^2x + sin^2x = 1
write the pythagorean property for cosine and sine
cos^2x + sin^2x = 1
cos (x + x) = ___?___ in terms of cos x and sin x
cos^2x - sin^2x
cos 2x = cos (x + x) = ___?___ in terms of cosine and sines of x
cos^2x - sin^2x
if θ is in standard position, then horizontal coordinate/radius is the definition of ___?___
cosine
which trigonometric functions are even functions?
cosine and secant
state the quotient property for cotangent in terms of sine and cosine
cosx/sinx
the equation 3(x + y) = 3x + 3y is an example of the ___?___ property for multiplication over addition
distributive
the graph of the parametric function x = 5 cos t and y = 4 sin t is a(n) ___?___
ellipse
what geometric figure is the graph of the parametric functions x = 3 cos t and y = 5 sin t
ellipse
x = 3 + 2 cos t and y = 5 + 4 sin t are parametric equations for a(n) ___?___
ellipse
is f(x)= 2^x an exponential function or a power function?
exponential
the equation y = 3 • 5^x represents a particular ___?___function
exponential
y = 3(1.06^x) is an example of a(n) ___?___ function
exponential
what kind of function is y = 5^x
exponential function
if f(x)= 2x-3, then f^-1(x)= ?
f^-1(x)= (x+3)/2
if f(x) = 2x + 6, then f^-1(x)= ?
f^-1(x)= (x-6)/2
if f(x)= x^2, write the equation for the inverse relation
f^-1(x)= +/- root x
if f(x)= x-3, then f^-1(x)= ?
f^-1(x)= x+3
if f(x)= 2x then f^-1(x)= ?
f^-1(x)= x/2
in triangle FED, the law of cosines states that f^2 = ___?___
f^2 = e^2 + d^2 - 2(e)(d) cos F
true or false: tan (x + y) = tan x + tan y
false
write the definition of a one to one function
function f is one to one if there are no y values that correspond to more than one x value
if f(x)= 2^x, find an equation for g(x), a horizontal translation of f(x) by -3 units
g(x)= f(x+3)
if g is a horizontal translation of f by -4 spaces, then g(x)= ___?___
g(x)= f(x+4)
finding equations of two sinusoids that are combined to form a graph is called ___?___
harmonic analysis
refer to right triangle QUI where U is the right angle and Q is the top angle; in terms of sides u and q, what does i equal?
i = root u^2 - q^2
refer to right triangle QUI where U is the right angle and Q is the top angle; in terms of side u and angle I, what does i equal?
i = u sin I
refer to right triangle QUI where U is the right angle and Q is the top angle; tan I = ___?___
i/q
refer to right triangle QUI where U is the right angle and Q is the top angle; cos Q = ___?___
i/u
for what value of x will the graph of y=x-3/x-5 have a discontinuity?
if x=5 it will have a discontinuity
in the composite function m(d(x)), function d is called the ___?___ function.
inside
if the inverse relation for function f is also a function, then f is called ___?___
invertible
explain why f^-1(x)= +/- root x is not a function
it does not pass the vertical line test (see 1-6)
the largest angle in a triangle is opposite the ___?___ side
largest
refer to triangle ABC: A = 38°, b = 4, c = 7; what method do you use to find the length of the third side?
law of cosines
log x + log y = log (___?___)
log (x•y)
log 3 + log 4 = log ___?___
log 12
for triangle MNO, sin M = 0.12, sin N = 0.3, and side m = 24 cm. how long is side n?
n = 60 cm
what is the first positive value of θ for which csc θ = 0
never
is cos x sin x = 1 an identity?
no
a(n) ___?___ triangle has no right angle
oblique
in the composite function m(d(x)), function m is called the ___?___ function.
outside
if p is a vertical dilation of f by a factor of .2 then p(x)= ___?___
p(x)= .2f(x)
state the law of cosines for triangle PAF involving angle P
p^2 = a^2 + f^2 - 2(a)(f) cos P
the graph of y = 3x^2 + 2x - 7 is called a(n)....
parabola
a function that repeats its values at regular intervals is called a ____?____ function
periodic
functions that repeat themselves at regular intervals are called ___?___ functions
periodic
the equation of y=3x^1.2 represents a particular ___?___ function
power
what kind of function is y = x^5
power function
y= 3x^2 + 5x -7 is a particular example of a _____?_____ function.
quadratic function
tan x = secx/cscx is called a ___?___ property
quotient
on a (u,v) coordinate plane (a right triangle is drawn): which side is the hypotenuse?
r
refer to a (u,v) coordinate plane (right triangle): sec θ
r/u
refer to a (u,v) coordinate plane (right triangle): cot θ
r/v
refer to a (u,v) coordinate plane (right triangle): csc θ
r/v
refer to triangle RTS; state the law of cosines using angle R
r^2 = t^2 + s^2 - 2(t)(s) cos R
how do you tell that sine is an odd function?
reflection of itself through the origin
find the amplitude of the sinusoid y = 2 cos θ + 5 sin θ
root 29
what is the exact value of tan 60°
root 3
what is the exact value of tan 60°?
root 3
find the exact value of sin 60°
root 3/2
sin 60°
root 3/2
what is the exact value of cos (π/6)
root 3/2
what is the exact value of cos 30°
root 3/2
what is the exact value of cos 30°?
root 3/2
refer to triangle RTS; state the law of cosines using angle S
s^2 = t^2 + r^2 - 2(t)(r) cos S
a(n) ___?___ triangle has no equal sides and no equal angles
scalene
write the reciprocal property for secant
secx = 1/cosx
sketch the graph of the relation that is not a function
see 1-3 if confused
sketch the graph of y= 2/3x + 4
see 1-3 if confused
sketch the graph of y=2^x
see 2-2
sketch a graph showing the distance of your foot from the pavement as a function of the distance your bicycle travels
see 2-3
sketch the graph of y= -x^2
see 2-4
sketch the graph of y = sec θ
see 3-4
sketch the graph of y = tan θ
see 3-4
sketch the graph of y = csc θ
see 3-6
sketch the graph of y = sin θ
see 3-6
sketch the graph of the parent cosine function y = cos x
see 4-3
sketch the graph of the parent sine function y = sin θ
see 4-3
sketch a reasonable graph for the time of sunset as a function of the day of the year
see 4-4
sketch the graph of y = sec x
see 4-4
sketch the reference angle for 260°
see 4-4
sketch the graph of an exponential function with base between 0 and 1
see 5-3
sketch the graph of the parent circular sinusoid y = sin x
see 5-5
sketch the graph of the parent trigonometric sinusoid y = cos θ
see 5-5
draw a sketch showing a vector sum
see 6-7
draw a sketch showing the components of vector v
see 6-7
sketch triangle XYZ, given x, y and angle X, showing how you can draw two possible triangles
see 6-7
the acronym SAS stands for ___?___
side angle side
by the composite argument properties, sin x cos y - cos x sin y = ___?___
sin (x - y)
refer to cos 57° and sin 33° equaling the same number; what relationship exists between the values of sine and cosine?
sin (x) = cos (90°-x)
express sin 2x in terms of sin x and cos x
sin 2x = 2sinxcosx
sin 5 cos 3 + cos 5 sin 3 = sin (___?___)
sin 8
by the composite argument properties, sin (A - B) = ___?___
sin A cos B - cos A sin B
for triangle ABC, write the law of sines involving angles A and C
sin A/a = sin C/c
refer to the equation: y = 3 + 4 cos 5 (θ - 6°); the graph of the equation is called a ___?___
sinusoid
the graph of the periodic function y= cosθ is called a ___?___
sinusoid
the graph of y - 5 cos θ + sin 12 θ is periodic with a varying ___?___
sinusoidal axis
why does (tanx)(cotx) = 1
sinx/cosx x cosx/sinx = 1
what is the first step in proving that a trigonometric equation is an identity?
starting with the more complex side and writing "proof"
refer to triangle RTS; state the law of cosines using angle T
t^2 = r^2 + s^2 - 2(r)(s) cos T
write the quotient property for tangent in terms of secant and cosecant
tanx = secx/cscx
write the quotient property for tangent in terms of sine and cosine
tanx = sinx/cosx
if the two legs of a right triangle are 57 and 65, find the tangent of the smallest angle
tanθ = 57/65
sketch an isosceles triangle
two sides equal length (see 1-3)
on a (u,v) coordinate plane (a right triangle is drawn): which side is the leg adjacent to θ?
u
cos θ = ?
u/r
refer to a (u,v) coordinate plane (right triangle): cos θ
u/r
tan 90°
undefined
on a (u,v) coordinate plane (a right triangle is drawn): which side is the leg opposite of θ?
v
refer to a (u,v) coordinate plane (right triangle): sin θ
v/r
refer to a (u,v) coordinate plane (right triangle): tan θ
v/r
sin θ = ?
v/r
find values of x and y if x + y = 20 and x - y = 12
x = 16, y = 4
factor: x^2 - 5x -6
x = 6, -1
given A = arcsin x, write the general solution for A in terms of sin^-1 x
x = sin^-1 (A) + 2πn or x = (π - sin^-1 (A)) + 2πn
calculate the product: (x-7)(x+8)
x^2 + x - 56
simplify: x^3x^5
x^8
refer to the equation: y = 3 + 4 cos 5 (θ - 6°); lower bound?
y = -1
refer to the equation: y = 3 + 4 cos 5 (θ - 6°); sinusoidal axis?
y = 3
refer to y = 4 + 5 cos π/6 (x - 7); where is the sinusoidal axis?
y = 4
write the particular equation of the sinusoid with amplitude 2, period 120°, sinusoidal axis at y = 5, and phase displacement 17° (for cosine)
y = 5 + 2 cos 3 (θ - 17°)
find the upper bound of y = 3 + 4 cos 5 (θ - 6°)
y = 7
refer to y = 4 + 5 cos π/6 (x - 7); where is the upper bound?
y = 9
what is the general equation of an exponential function?
y = ab^x
write the general equation of a quadratic function
y = ax^2 + bx + c
right triangle XYZ has right angle Y. side x is opposite angle X, and so on. find csc X
y/x
what is the transformation of a cosine graph that has a vertical dilation of 4
y= 4 cos θ
write an equation for a cosine graph with a vertical dilation of 4
y= 4 cos θ
write the general equation of a quadratic function (#1)
y= ax^2 + bx + c
write the general equation of a power function
y= ax^b
write an equation for the parent sinusoid graph
y= sin θ
write the general equation of an exponential function
y=ab^x
write the general equation of an exponential function (#2)
y=ab^x
is cos^2x = 1 - sin^2x and identity?
yes
is cot x tan x = 1 an identity?
yes
why do you need only the function cos^-1, not the relation arccos, when using the law of cosines?
you do not need to apply the relation arccos because you only need the positive cos^-1 and an angle measure between 0 and 180 degrees since the angle always needs to fit within the triangle
find the smaller acute angle of a right triangle with legs of 3 miles and 7 miles
θ = 23.1985...°
find the measure of the smaller acute angle of a right triangle with legs 13 and 28 cm
θ = 24.9047...°
find the measure of the larger acute angle of a right triangle with legs 11 ft and 9 ft
θ = 50.7105...°
find the phase displacement for y = cos x of y = 2 cos θ + 5 sin θ
θ = 68.1985...°
if θ = csc^-1 (11/7), then θ = sin^-1 (___?___)
θ = sin^-1 (7/11)
the reference angle for 260° is ___?___
θ ref = 80°
how many radians are there in 180°
π
how many radians in 180°
π
what is the period of the circular function y = cos 4x
π/2
two values of x = arccos .5 are ___?___ and ___?___
π/3 and -π/3
how many radians in 45°?
π/4
the value of the inverse circular function x = sin^-1 (0.5) is ___?___
π/6
if cos A = 0.6, sin A = 0.8, cos B = 1/root 2, and sin B = -1/root 2, then cos (A - B) = ___?___
(-.2/root 2)
give another symbol for m(d(x))
(m x d) (x)
refer to triangle RTS; express cos T in terms of sides r, s, and t
(t^2 - r^2 - s^2)/-2rs = cos T
if cos^-1x = 1.2, what is the general solution for arccos x?
+/- 1.2 + 2πn
if sin θ = 0.372..., then sin (-θ) = ___?___
-0.372... (because sin (-θ) = -sin θ)
evaluate cos π
-1
tan^2 47° - sec^2 47° = ___?___
-1
cos 135°
-1/root 2
write vector a + vector b if vector a = 4i + 7j and vector b = -6i + 8j
-2i + 15j
what is the phase displacement for y = 7 + 6 cos 5 (θ + 37°) with respect to the parent cosine function?
-37° (cos)
write 5% as a decimal
.05
find sin 47
.1235...
write cos 57° in decimal form
.5446...
write sin 33° in decimal form
.5446...
how many radians in 34°
.5934...
find sin 47°
.7313...
what is the exact value of cot (π/2)
0
3.7^0 = ?
1
if f(x)= 2^x, find f(0)
1
refer to right triangle QUI where U is the right angle and Q is the top angle; sin U = ___?___
1
state the pythagorean property for secant and tangent
1 + tan^2x = sec^2x
what is the pythagorean property for secant and tangent?
1 + tan^2x = sec^2x
write the pythagorean property that involves tangent
1 + tan^2x = sec^2x
refer to y = 4 + 5 cos π/6 (x - 7); the first three positive x-values at which low points occur are ___?___, ___?___, and ___?___
1, 13, 25
refer to y = 4 + 5 cos π/6 (x - 7); what is the frequency?
1/12
the function y = 5 + 6 cos 7 (x - 8) is a horizontal dilation of y = cos x by ___?___
1/7
refer to the equation: y = 3 + 4 cos 5 (θ - 6°); frequency?
1/72
find the exact value of cos π/4
1/root 2
what is the exact value of sin (π/4)
1/root 2
what is the exact value of sin (π/4)?
1/root 2
find the exact value of tan 30°
1/root 3
what is the exact value of tan 30°
1/root 3
state the reciprocal property for cosecant
1/sinx
2^10 = ?
1024
refer to y = 4 + 5 cos π/6 (x - 7); what is the period?
12
find 40% of 300
120
what is the value of 5!
120
find the approximate value of cot^-1 (4.3)
13.0918...
if one value of arcsin x is 30°, find another positive value of arcsin x, less than 360°
150°
what is the amplitude of the sinusoid y = 8 cos θ + 15 sin θ?
17
find the degree measure of the acute angle cot^-1 (3)
18.4349°...
the sum of the angle measures in a triangle is ___?___
180°
what is the first positive value of θ at which y = cot θ has a vertical asymptote?
180°
if y = cosBθ has a period of 180°, what does B equal?
2
|3-5| = ?
2
refer to fig. 3-2l on pg. 92; how many cycles are there between θ= 20° and θ=80°
2 cycles
what percent of 300 is 60
20%
40 is 20% of what number?
200
what is the period of the parent sine function y = sin x
2π
what is the period of the circular function y = sin 5x
2π/5
what is the period of the function y = 5 + 6 cos 7 (x - 8)
2π/7
what is the vertical dilation for y = 2 + 3 cos 4 (x - 5)
3
if f(x)= 2^x then f^-1(8)= ?
3 (see 1-6)
the value of 30 is what percentage of 1000?
3%
find the approximate value of sec 71°
3.0715...
how many degrees in π/6 radians?
30°
if an angle has a measure of π/6 radians, what is its degree measure?
30°
refer to fig. 3-2l on pg. 92; what is the period?
30°
write two values of θ = sin^-1 0.5 that lie between 0° and 180°
30° and 150°
find x if 5 log 2 = log x
32
how many degrees in 2π radians?
360°
if f(x) = x^3, then the inverse function f^-1(x) = ___?___
3root x
if f(x)= 2x and g(x)= x+3, find f(f(1))
4
refer to the equation: y = 3 + 4 cos 5 (θ - 6°); amplitude?
4
the amplitude of y = 3 + 4 cos 5 (x - 6) is ___?___
4
how many degrees are there in π/4 radians
45°
find the amplitude of the sinusoid y = 4 cos x + 3 sin x
5
if f(x)= 2x and g(x)= x+3, find g(f(1))
5
refer to y = 4 + 5 cos π/6 (x - 7); what is the amplitude?
5
the function y = 5 + 6 cos 7 (x - 8) is a vertical translation of y = cos x by ___?___
5
what is the value of n if log 32 = n log 2
5
how long does it take you to go 300 miles at an average speed of 60 mi/h?
5 hours
if sin^-1x = 56°, what is the general solution for arcsin x?
56° + 360°n or (180°-56°) + 360°n
how many degrees in 1 radian?
57.2957...°
a value of the inverse circular relation x = arcsin 0.5 between π/2 and 2π is ___?___
5π/6
the function y = 5 + 6 cos 7 (x - 8) is a vertical dilation of y = cos x by ___?___
6
write csc 9° in decimal form
6.3924
write sec 81° in decimal form
6.3924...
refer to y = 4 + 5 cos π/6 (x - 7); if x = 9, then y = ___?___
6.5
find the first three positive angles of θ = arccos .5
60°, 300°, 420°
find the reference angle for 241 deg
61 deg
if A cos (θ - D) = 8 cos θ + 15 sin θ, then D could equal ___?___
61.9275...°
find in decimal degrees θ = cot^-1 (3/7)
66.8014...°
refer to the equation: y = 3 + 4 cos 5 (θ - 6°); phase displacement?
6° (cos)
refer to y = 4 + 5 cos π/6 (x - 7); what is the phase displacement?
7 (cos)
how many degrees are there in 2 revolutions?
720
find the period of y = 3 + 4 cos 5 (θ - 6°)
72°
refer to the equation: y = 3 + 4 cos 5 (θ - 6°); period?
72°
if f(x)= 2x and g(x)= x+3, find f(g(1))
8
refer to fig. 3-2l on pg. 92; what is the amplitude?
8
the function y = 5 + 6 cos 7 (x - 8) is a horizontal translation of y = cos x by ___?___
8
the period of the sinusoid y = 5 + 7 cos π/4 (x - 6) is ___?___
8
3^2005/3^2001 = ___?___
81
if y = 5 x 3^x, then adding 2 to the value of x multiplies the value of y by ___?___
9
find 30% of 3000
900
what is the period of the trigonometric function y = cos 4θ
90°
expand: (3x-5)^2
9x^2 - 30x + 25
find values of x and y if x + y = A and x - y = B
A = (x + y)/2, B = (x - y)/2
state the formula for the area of triangle PAF involving angle P
A = 1/2 (a) (f) sin P
for triangle ABC, write the area formula involving angle A
A = 1/2 (b) (c) sin A
a triangle has sides 5 ft and 8 ft and included angle 30°. what is the area of the triangle?
A = 10
refer to triangle ABC: A = 38°, b = 4, c = 7; find the area
A = 8.6192... units^2
tan π/4 = a. 1 b. 0 c. -1 d. 1/2 e. root 3/2
A. 1
the period of the circular function y = 3 + 7 cos (π/8) (x - 1) is a. 16 b. 8 c. π/8 d. 7 e. 3
A. 16
x^20/x^5= a. x^15 d. x^100 b. x^4 e. none c. x^25
A. x^15
cos 90° = a. 1 b. 0 c. -1 d. 1/2 e. root 3/2
B. 0
the exact value of cos π/4 is... a. 0 b. 1/root 2 c. 1/2 d. root 3/2 e. 1
B. 1/root 2
the "if..." part if the statement of a theorem is called the... a. conclusion b. hypothesis c. converse d. inverse e. contrapositive
B. hypothesis
cos π = a. 1 b. 0 c. -1 d. 1/2 e. root 3/2
C. -1
which of these is a horizontal dilation by a factor of 2? a. g(x)= 2f(x) c. g(x)= f(0.5x) b. g(x)= 0.5f(x) d. g(x)= f(2x)
C. g(x)= f(0.5x)
the "then" part of the statement of a theorem is called the... a. converse b. inverse c. contrapositive d. conclusion e. hypothesis
D. conclusion
cos (π/6) = a. 1/root 3 b. 1/2 c. 2/root 3 d. root 3/2 e. root 3
D. root 3/2
which one of these is not the equation of a function? a. y=3x + 5 b. f(x)= 3 - x^2 c. g(x)= |x| d. y= +/- root x e. y= 5x^2/3
D. y= +/- root x
a one to one function is... a. always increasing b. always decreasing c. always positive d. always negative e. always invertible
E. always invertible
the period of the circular function y = 3 + 4 cos 5 (x - 6) is... a. 3 b. 4 c. 5 d. 6 e. none
E. none
what transformation of function f is represented by h(x) = f(10x)
HD of 1/10
describe the transformation: g(x)= f(3x)
HD of 1/3
what transformation is applied to f(x) to get g(x) = f(3x)
HD of 1/3
what transformation of y = cos x is expressed by y = cos 5x
HD of 1/5
refer to fig. 3-2l on pg. 92; what is the horizontal translation for cosine (phase displacement)?
HT (cos) of 20°
describe the transformation: g(x)= f(x-2)
HT of 2
what transformation is the image function y=(x-3)^5 of the pre-image y=x^5
HT of 3
if g(x) = f (x - 7), what transformation is done on function f to get function g?
HT of 7
refer to right triangle QUI where U is the right angle and Q is the top angle; in terms of the inverse tangent function, angle Q = ___?___
Q = tan^-1 (q/i)