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Math Problems
Algebra 2
Write equations of cosine functions using properties
A complex number
z
1
z_{1}
z
1
has a magnitude
∣
z
1
∣
=
20
\left|z_{1}\right|=20
∣
z
1
∣
=
20
and an angle
θ
1
=
28
1
∘
\theta_{1}=281^{\circ}
θ
1
=
28
1
∘
.
\newline
Express
z
1
z_{1}
z
1
in rectangular form, as
z
1
=
a
+
b
i
z_{1}=a+b i
z
1
=
a
+
bi
.
\newline
Round
a
a
a
and
b
b
b
to the nearest thousandth.
\newline
z
1
=
□
+
□
i
z_{1}=\square+\square i
z
1
=
□
+
□
i
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A complex number
z
1
z_{1}
z
1
has a magnitude
∣
z
1
∣
=
2
\left|z_{1}\right|=2
∣
z
1
∣
=
2
and an angle
θ
1
=
4
9
∘
\theta_{1}=49^{\circ}
θ
1
=
4
9
∘
.
\newline
Express
z
1
z_{1}
z
1
in rectangular form, as
z
1
=
a
+
b
i
z_{1}=a+b i
z
1
=
a
+
bi
.
\newline
Round
a
a
a
and
b
b
b
to the nearest thousandth.
\newline
z
1
=
□
+
□
i
z_{1}=\square+\square i
z
1
=
□
+
□
i
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A complex number
z
1
z_{1}
z
1
has a magnitude
∣
z
1
∣
=
10
\left|z_{1}\right|=10
∣
z
1
∣
=
10
and an angle
θ
1
=
8
4
∘
\theta_{1}=84^{\circ}
θ
1
=
8
4
∘
.
\newline
Express
z
1
z_{1}
z
1
in rectangular form, as
z
1
=
a
+
b
i
z_{1}=a+b i
z
1
=
a
+
bi
.
\newline
Round
a
a
a
and
b
b
b
to the nearest thousandth.
\newline
z
1
=
□
+
□
i
z_{1}=\square+\square i
z
1
=
□
+
□
i
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Vector
A
B
undefined
\overrightarrow{A B}
A
B
has an initial point
(
1
,
−
4
)
(1,-4)
(
1
,
−
4
)
, an
x
x
x
component of
−
8
-8
−
8
, and a
y
y
y
component of
9
9
9
.
\newline
Find the coordinates of the terminal point,
B
B
B
.
\newline
B
=
(
□
,
□
)
B=(\square, \square)
B
=
(
□
,
□
)
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Vector
A
B
undefined
\overrightarrow{A B}
A
B
has a terminal point
(
−
3
,
−
6
)
(-3,-6)
(
−
3
,
−
6
)
, an
x
x
x
component of
−
6
-6
−
6
, and a
y
y
y
component of
−
8
-8
−
8
.
\newline
Find the coordinates of the initial point,
A
A
A
.
\newline
A
=
(
□
,
□
)
A=(\square, \square)
A
=
(
□
,
□
)
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Vector
A
B
undefined
\overrightarrow{A B}
A
B
has an initial point
(
7
,
−
4
)
(7,-4)
(
7
,
−
4
)
, an
x
x
x
component of
−
10
-10
−
10
, and a
y
y
y
component of
−
5
-5
−
5
.
\newline
Find the coordinates of the terminal point,
B
B
B
.
\newline
B
=
(
□
,
□
)
B=(\square, \square)
B
=
(
□
,
□
)
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Vector
A
B
undefined
\overrightarrow{A B}
A
B
has an initial point
(
−
9
,
8
)
(-9,8)
(
−
9
,
8
)
, an
x
x
x
component of
4
4
4
, and a
y
y
y
component of
−
11
-11
−
11
.
\newline
Find the coordinates of the terminal point,
B
B
B
.
\newline
B
=
(
□
,
□
)
B=(\square, \square)
B
=
(
□
,
□
)
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Vector
A
B
undefined
\overrightarrow{A B}
A
B
has a terminal point
(
4
,
−
7
)
(4,-7)
(
4
,
−
7
)
, an
x
x
x
component of
−
3
-3
−
3
, and a
y
y
y
component of
−
9
-9
−
9
.
\newline
Find the coordinates of the initial point,
A
A
A
.
\newline
A
=
(
□
,
□
)
A=(\square, \square)
A
=
(
□
,
□
)
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Vector
A
B
undefined
\overrightarrow{A B}
A
B
has an initial point
(
3
,
4
)
(3,4)
(
3
,
4
)
, an
x
x
x
component of
5
5
5
, and a
y
y
y
component of
4
4
4
.
\newline
Find the coordinates of the terminal point,
B
B
B
.
\newline
B
=
(
□
,
□
)
B=(\square, \square)
B
=
(
□
,
□
)
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Vector
A
B
undefined
\overrightarrow{A B}
A
B
has a terminal point
(
7
,
−
9
)
(7,-9)
(
7
,
−
9
)
, an
x
x
x
component of
13
13
13
, and a
y
y
y
component of
−
5
-5
−
5
.
\newline
Find the coordinates of the initial point,
A
A
A
.
\newline
A
=
(
□
,
□
)
A=(\square, \square)
A
=
(
□
,
□
)
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Vector
A
B
undefined
\overrightarrow{A B}
A
B
has a terminal point
(
7
,
9
)
(7,9)
(
7
,
9
)
, an
x
x
x
component of
11
11
11
, and a
y
y
y
component of
12
12
12
.
\newline
Find the coordinates of the initial point,
A
A
A
.
\newline
A
=
(
□
,
□
)
A=(\square, \square)
A
=
(
□
,
□
)
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A toy boat is bobbing on the water.
\newline
Its distance
D
(
t
)
D(t)
D
(
t
)
(in
m
\mathrm{m}
m
) from the floor of the lake as a function of time
t
t
t
(in seconds) can be modeled by a sinusoidal expression of the form
a
⋅
sin
(
b
⋅
t
)
+
d
a \cdot \sin (b \cdot t)+d
a
⋅
sin
(
b
⋅
t
)
+
d
.
\newline
At
t
=
0
t=0
t
=
0
, when the boat is exactly in the middle of its oscillation, it is
1
m
1 \mathrm{~m}
1
m
above the water's floor. The boat reaches its maximum height of
1.2
m
1.2 \mathrm{~m}
1.2
m
after
π
4
\frac{\pi}{4}
4
π
seconds.
\newline
Find
D
(
t
)
D(t)
D
(
t
)
.
\newline
t
t
t
should be in radians.
\newline
D
(
t
)
=
□
D(t)=\square
D
(
t
)
=
□
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The following formula gives the surface area
S
S
S
of a right cylinder, where
r
r
r
is the radius of the base and
h
h
h
is the height:
\newline
S
=
2
π
r
(
r
+
h
)
S=2 \pi r(r+h)
S
=
2
π
r
(
r
+
h
)
\newline
Rearrange the formula to highlight the height.
\newline
h
=
□
h=\square
h
=
□
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The graph of a sinusoidal function intersects its midline at
(
0
,
−
6
)
(0,-6)
(
0
,
−
6
)
and then has a minimum point at
(
2.5
,
−
9
)
(2.5,-9)
(
2.5
,
−
9
)
.
\newline
Write the formula of the function, where
x
x
x
is entered in radians.
\newline
f
(
x
)
=
□
f(x)=\square
f
(
x
)
=
□
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The graph of a sinusoidal function intersects its midline at
(
0
,
−
2
)
(0,-2)
(
0
,
−
2
)
and then has a minimum point at
(
3
π
2
,
−
7
)
\left(\frac{3 \pi}{2},-7\right)
(
2
3
π
,
−
7
)
.
\newline
Write the formula of the function, where
x
x
x
is entered in radians.
\newline
f
(
x
)
=
f(x)=
f
(
x
)
=
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The graph of a sinusoidal function intersects its midline at
(
0
,
1
)
(0,1)
(
0
,
1
)
and then has a maximum point at
(
7
π
4
,
5
)
\left(\frac{7 \pi}{4}, 5\right)
(
4
7
π
,
5
)
.
\newline
Write the formula of the function, where
x
x
x
is entered in radians.
\newline
f
(
x
)
=
□
f(x)=\square
f
(
x
)
=
□
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The graph of a sinusoidal function has a maximum point at
(
0
,
10
)
(0,10)
(
0
,
10
)
and then intersects its midline at
(
π
4
,
4
)
\left(\frac{\pi}{4}, 4\right)
(
4
π
,
4
)
.
\newline
Write the formula of the function, where
x
x
x
is entered in radians.
\newline
f
(
x
)
=
f(x)=
f
(
x
)
=
Get tutor help
The graph of a sinusoidal function has a maximum point at
(
0
,
5
)
(0,5)
(
0
,
5
)
and then has a minimum point at
(
2
π
,
−
5
)
(2 \pi,-5)
(
2
π
,
−
5
)
.
\newline
Write the formula of the function, where
x
x
x
is entered in radians.
\newline
f
(
x
)
=
f(x)=
f
(
x
)
=
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x
(
x
−
a
)
−
b
(
a
+
b
)
=
0
x(x-a)-b(a+b)=0
x
(
x
−
a
)
−
b
(
a
+
b
)
=
0
\newline
In the given equation,
a
a
a
and
b
b
b
are constants. What are the solutions to the equation?
\newline
Choose
1
1
1
answer:
\newline
(A)
x
=
−
b
x=-b
x
=
−
b
and
x
=
(
a
−
b
)
x=(a-b)
x
=
(
a
−
b
)
\newline
(B)
x
=
−
b
x=-b
x
=
−
b
and
x
=
(
a
+
b
)
x=(a+b)
x
=
(
a
+
b
)
\newline
(C)
x
=
a
x=a
x
=
a
and
x
=
(
a
−
b
)
x=(a-b)
x
=
(
a
−
b
)
\newline
(D)
x
=
a
x=a
x
=
a
and
x
=
(
a
+
b
)
x=(a+b)
x
=
(
a
+
b
)
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(
3
x
2
−
7
x
)
+
(
5
x
2
+
13
)
\left(3 x^{2}-7 x\right)+\left(5 x^{2}+13\right)
(
3
x
2
−
7
x
)
+
(
5
x
2
+
13
)
\newline
The given expression can be written as
a
x
2
+
b
x
+
c
a x^{2}+b x+c
a
x
2
+
b
x
+
c
, where
a
,
b
a, b
a
,
b
, and
c
c
c
are constants. What is the value of
a
a
a
?
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For a given input value
a
a
a
, the function
f
f
f
outputs a value
b
b
b
to satisfy the following equation.
\newline
−
3
a
+
6
b
=
a
+
4
b
-3a+6b=a+4b
−
3
a
+
6
b
=
a
+
4
b
\newline
Write a formula for
f
(
a
)
f(a)
f
(
a
)
in terms of
a
a
a
.
\newline
f
(
a
)
=
□
f(a)=\square
f
(
a
)
=
□
Get tutor help
For a given input value
x
x
x
, the function
g
g
g
outputs a value
y
y
y
to satisfy the following equation.
\newline
−
4
x
−
6
=
−
5
y
+
2
-4x-6=-5y+2
−
4
x
−
6
=
−
5
y
+
2
\newline
Write a formula for
g
(
x
)
g(x)
g
(
x
)
in terms of
x
x
x
.
\newline
g
(
x
)
=
□
g(x)=\square
g
(
x
)
=
□
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For a given input value
b
b
b
, the function
f
f
f
outputs a value
a
a
a
to satisfy the following equation.
\newline
4
a
+
7
b
=
−
52
4a+7b=-52
4
a
+
7
b
=
−
52
\newline
Write a formula for
f
(
b
)
f(b)
f
(
b
)
in terms of
b
b
b
.
\newline
f
(
b
)
=
□
f(b)=\square
f
(
b
)
=
□
Get tutor help
Kris is wrapping Christmas lights around the railing that runs around three sides of his square porch If one side of his porch is
12
12
12
feet long, and each strand of his Christmas lights will wrap around a
70
70
70
-inch length of the railing, what is the minimum number of strands Kris needs to completely wrap the porch railing?
\newline
A)
6
6
6
\newline
B)
7
7
7
\newline
C)
8
8
8
\newline
D)
9
9
9
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Kris is wrapping Christmas lights around the railing that runs around three sides of his square porch If one side of his porch is
12
12
12
feet long, and each strand of his Christmes lights will wrap around a
70
70
70
-inch length of the railing, what is the minimum number of strands Kris needs to completely wrap the porch railing?
\newline
A)
6
6
6
\newline
B)
7
7
7
\newline
C)
8
8
8
\newline
D)
9
9
9
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Find an equation for a sinusoidal function that has period
2
π
2\pi
2
π
, amplitude
1
1
1
, and contains the point
(
0
,
−
1
)
(0,-1)
(
0
,
−
1
)
.
\newline
Write your answer in the form
f
(
x
)
=
A
cos
(
B
x
+
C
)
+
D
f(x) = A\cos(Bx + C) + D
f
(
x
)
=
A
cos
(
B
x
+
C
)
+
D
, where
A
A
A
,
B
B
B
,
C
C
C
, and
D
D
D
are real numbers.
\newline
f
(
x
)
=
f(x) =
f
(
x
)
=
_____
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