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Math Problems
Precalculus
Find trigonometric ratios using a Pythagorean or reciprocal identity
Let
x
x
x
and
y
y
y
be functions of
t
t
t
with
y
=
15
cos
x
y = 15\cos x
y
=
15
cos
x
. If
d
x
d
t
=
−
1
9
\frac{dx}{dt} = -\frac{1}{9}
d
t
d
x
=
−
9
1
, what is
d
y
d
t
\frac{dy}{dt}
d
t
d
y
when
x
=
π
6
x = \frac{\pi}{6}
x
=
6
π
?
\newline
Write an exact, simplified answer.
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The trisectors of angles
B
B
B
and
C
C
C
of scalene triangle
A
B
C
A B C
A
BC
meet at points
P
P
P
and
Q
Q
Q
, as shown. Angle
A
A
A
measures
39
39
39
degrees and angle QBP measures
14
14
14
degrees. What is the measure of angle BPC?
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The planet Neptune orbits the Sun. Its orbital radius is
30
30
30
.
1
1
1
astronomical units (AU).
\newline
Assuming Neptune's orbit is circular, what is the distance it travels in a single orbit around the Sun?
\newline
Give your answer in terms of
π
\pi
π
.
\newline
□
\square
□
AU
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If
tan
(
x
+
20
)
=
cot
x
\tan(x + 20) = \cot x
tan
(
x
+
20
)
=
cot
x
, a value of
x
x
x
is
\newline
a.
35
35
35
\newline
b.
45
45
45
\newline
c.
55
55
55
\newline
d.
70
70
70
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In the figure,
O
O
O
is the center of the circle. The minor arc
A
B
undefined
\widehat{A B}
A
B
has a length of
4
π
4 \pi
4
π
and is
1
8
\frac{1}{8}
8
1
the circumference of the circle. If the area of the shaded region is
a
π
a \pi
aπ
, what is the value of
a
a
a
?
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For the rotation
−
15
2
∘
-152^{\circ}
−
15
2
∘
, find the coterminal angle from
0
∘
≤
θ
<
36
0
∘
0^{\circ} \leq \theta<360^{\circ}
0
∘
≤
θ
<
36
0
∘
, the quadrant, and the reference angle.
\newline
The coterminal angle is
□
∘
\square^{\circ}
□
∘
, which lies in Quadrant
□
\square
□
, with a reference angle of
□
∘
\square^{\circ}
□
∘
.
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For the rotation
−
105
3
∘
-1053^{\circ}
−
105
3
∘
, find the coterminal angle from
0
∘
≤
θ
<
36
0
∘
0^{\circ} \leq \theta<360^{\circ}
0
∘
≤
θ
<
36
0
∘
, the quadrant, and the reference angle.
\newline
The coterminal angle is
□
∘
\square^{\circ}
□
∘
, which lies in Quadrant
□
\square
□
, with a reference angle of
□
∘
\square^{\circ}
□
∘
.
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Given the vector
v
\mathbf{v}
v
has an initial point at
(
4
,
7
)
(4,7)
(
4
,
7
)
and a terminal point at
(
−
1
,
7
)
(-1,7)
(
−
1
,
7
)
, find the exact value of
∥
v
∥
\|\mathbf{v}\|
∥
v
∥
.
\newline
Answer:
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simplify this
r
=
C
p
C
v
=
C
v
+
R
C
v
=
R
r
+
1
+
R
R
/
r
+
1
r = \frac{C_p}{C_v} = \frac{C_v + R}{C_v} = \frac{R}{r} + 1 + \frac{R}{R/r} + 1
r
=
C
v
C
p
=
C
v
C
v
+
R
=
r
R
+
1
+
R
/
r
R
+
1
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The angle
θ
1
\theta_{1}
θ
1
is located in Quadrant III, and
sin
(
θ
1
)
=
−
12
13
\sin \left(\theta_{1}\right)=-\frac{12}{13}
sin
(
θ
1
)
=
−
13
12
.
\newline
What is the value of
cos
(
θ
1
)
\cos \left(\theta_{1}\right)
cos
(
θ
1
)
?
\newline
Express your answer exactly.
\newline
cos
(
θ
1
)
=
\cos \left(\theta_{1}\right)=
cos
(
θ
1
)
=
\newline
□
\square
□
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What is the exact value of
tan
(
19
π
12
)
\tan(\frac{19\pi}{12})
tan
(
12
19
π
)
?
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Given that
sin
x
=
24
6
\sin x=\frac{\sqrt{24}}{6}
sin
x
=
6
24
and
sin
y
=
5
3
\sin y=\frac{\sqrt{5}}{3}
sin
y
=
3
5
, and that angles
x
x
x
and
y
y
y
are both in Quadrant I, find the exact value of
sin
(
x
+
y
)
\sin (x+y)
sin
(
x
+
y
)
, in simplest radical form.
\newline
Answer:
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Given that
cos
A
=
2
3
\cos A=\frac{\sqrt{2}}{3}
cos
A
=
3
2
and
cos
B
=
30
6
\cos B=\frac{\sqrt{30}}{6}
cos
B
=
6
30
, and that angles
A
A
A
and
B
B
B
are both in Quadrant I, find the exact value of
sin
(
A
−
B
)
\sin (A-B)
sin
(
A
−
B
)
, in simplest radical form.
\newline
Answer:
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Given that
tan
A
=
1
2
\tan A=\frac{1}{2}
tan
A
=
2
1
and
sin
B
=
6
85
\sin B=\frac{6}{\sqrt{85}}
sin
B
=
85
6
, and that angles
A
A
A
and
B
B
B
are both in Quadrant I, find the exact value of
cos
(
A
+
B
)
\cos (A+B)
cos
(
A
+
B
)
, in simplest radical form.
\newline
Answer:
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Given that
cos
x
=
3
10
\cos x=\frac{3}{\sqrt{10}}
cos
x
=
10
3
and
tan
y
=
3
2
\tan y=\frac{3}{2}
tan
y
=
2
3
, and that angles
x
x
x
and
y
y
y
are both in Quadrant I, find the exact value of
sin
(
x
−
y
)
\sin (x-y)
sin
(
x
−
y
)
, in simplest radical form.
\newline
Answer:
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Given that
cos
A
=
20
5
\cos A=\frac{\sqrt{20}}{5}
cos
A
=
5
20
and
cos
B
=
6
3
\cos B=\frac{\sqrt{6}}{3}
cos
B
=
3
6
, and that angles
A
A
A
and
B
B
B
are both in Quadrant I, find the exact value of
sin
(
A
−
B
)
\sin (A-B)
sin
(
A
−
B
)
, in simplest radical form.
\newline
Answer:
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Given that
tan
x
=
3
\tan x=\sqrt{3}
tan
x
=
3
and
cos
y
=
2
2
\cos y=\frac{\sqrt{2}}{2}
cos
y
=
2
2
, and that angles
x
x
x
and
y
y
y
are both in Quadrant I, find the exact value of
cos
(
x
+
y
)
\cos (x+y)
cos
(
x
+
y
)
, in simplest radical form.
\newline
Answer:
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Given the reference angle of
2
π
5
\frac{2 \pi}{5}
5
2
π
, find the corresponding angle in Quadrant
2
2
2
.
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
2
ln
(
2
x
−
2
)
+
12
=
8
2 \ln (2 x-2)+12=8
2
ln
(
2
x
−
2
)
+
12
=
8
\newline
Answer:
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Q
7
7
7
\newline
D
(
4.5
,
3
)
,
E
(
5
,
11
)
D(4.5,3), E(5,11)
D
(
4.5
,
3
)
,
E
(
5
,
11
)
and
F
(
2.5
,
5
)
F(2.5,5)
F
(
2.5
,
5
)
are three points in the
x
y
x y
x
y
-plane. If
D
F
‾
\overline{D F}
D
F
is a diameter of Circle
1
1
1
and
E
F
‾
\overline{E F}
EF
is a diameter of Circle
2
2
2
, what is the slope of the line that goes through the centers of the two circles?
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\newline
The angle
θ
1
\theta_{1}
θ
1
is located in Quadrant
I
\mathrm{I}
I
, and
sin
(
θ
1
)
=
17
20
\sin \left(\theta_{1}\right)=\frac{17}{20}
sin
(
θ
1
)
=
20
17
.
\newline
What is the value of
cos
(
θ
1
)
\cos \left(\theta_{1}\right)
cos
(
θ
1
)
?
\newline
Express your answer exactly.
\newline
cos
(
θ
1
)
=
\cos \left(\theta_{1}\right)=
cos
(
θ
1
)
=
Get tutor help
If
cos
(
θ
)
=
8
17
\cos(\theta) = \frac{8}{17}
cos
(
θ
)
=
17
8
and
0
∘
<
θ
<
9
0
∘
0^\circ < \theta < 90^\circ
0
∘
<
θ
<
9
0
∘
, what is
sec
(
θ
)
\sec(\theta)
sec
(
θ
)
?
\newline
Write your answer in simplified, rationalized form.
\newline
sec
(
θ
)
=
\sec(\theta) =
sec
(
θ
)
=
______
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Given
sec
A
=
61
6
\sec A = \frac{\sqrt{61}}{6}
sec
A
=
6
61
and that angle
A
A
A
is in Quadrant I, find the exact value of
csc
A
\csc A
csc
A
in simplest radical form using a rational denominator.
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E
=
[
0
3
5
5
5
2
]
\mathrm{E}=\left[\begin{array}{lll}0 & 3 & 5 \\ 5 & 5 & 2\end{array}\right]
E
=
[
0
5
3
5
5
2
]
and
D
=
[
3
4
3
−
2
4
−
2
]
\mathrm{D}=\left[\begin{array}{rr}3 & 4 \\ 3 & -2 \\ 4 & -2\end{array}\right]
D
=
⎣
⎡
3
3
4
4
−
2
−
2
⎦
⎤
\newline
Let
H
=
E
D
\mathrm{H}=\mathrm{ED}
H
=
ED
. Find
H
\mathrm{H}
H
.
\newline
H
=
\mathbf{H}=
H
=
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Express
z
1
=
6
+
2
3
i
z_{1}=6+2 \sqrt{3} i
z
1
=
6
+
2
3
i
in polar form.
\newline
Express your answer in exact terms, using degrees, where your angle is between
0
∘
0^{\circ}
0
∘
and
36
0
∘
360^{\circ}
36
0
∘
, inclusive.
\newline
z
1
=
z_{1}=
z
1
=
Get tutor help
Find the midpoint
m
m
m
of
z
1
=
(
7
+
8
i
)
z_{1}=(7+8 i)
z
1
=
(
7
+
8
i
)
and
z
2
=
(
8
−
7
i
)
z_{2}=(8-7 i)
z
2
=
(
8
−
7
i
)
.
\newline
Express your answer in rectangular form.
\newline
m
=
□
m=\square
m
=
□
Get tutor help
Express
z
1
=
−
8
3
+
8
i
z_{1}=-8 \sqrt{3}+8 i
z
1
=
−
8
3
+
8
i
in polar form.
\newline
Express your answer in exact terms, using degrees, where your angle is between
0
∘
0^{\circ}
0
∘
and
36
0
∘
360^{\circ}
36
0
∘
, inclusive.
\newline
z
1
=
z_{1}=
z
1
=
Get tutor help
z
=
−
8
+
5
i
z=-8+5 i
z
=
−
8
+
5
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
−
2
−
5
i
z=-2-5 i
z
=
−
2
−
5
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
−
3
+
5
i
z=-3+5 i
z
=
−
3
+
5
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
1
−
2
i
z=1-2 i
z
=
1
−
2
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
1
+
4
i
z=1+4 i
z
=
1
+
4
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
3
−
4
i
z=3-4 i
z
=
3
−
4
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
−
6
−
5
i
z=-6-5 i
z
=
−
6
−
5
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
8
−
9
i
z=8-9 i
z
=
8
−
9
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
4
+
3
i
z=4+3 i
z
=
4
+
3
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
−
5
+
3
i
z=-5+3 i
z
=
−
5
+
3
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
−
7
−
6
i
z=-7-6 i
z
=
−
7
−
6
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
−
3
−
8
i
z=-3-8 i
z
=
−
3
−
8
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
7
+
3
i
z=7+3 i
z
=
7
+
3
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
z
=
5
−
3
i
z=5-3 i
z
=
5
−
3
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
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z
=
6
+
5
i
z=6+5 i
z
=
6
+
5
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
5
+
7
i
z=5+7 i
z
=
5
+
7
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
−
8
+
3
i
z=-8+3 i
z
=
−
8
+
3
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
−
2
+
8
i
z=-2+8 i
z
=
−
2
+
8
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
z
=
4
−
2
i
z=4-2 i
z
=
4
−
2
i
\newline
Find the angle
θ
\theta
θ
(in degrees) that
z
z
z
makes in the complex plane.
\newline
Round your answer, if necessary, to the nearest tenth. Express
θ
\theta
θ
between
−
18
0
∘
-180^{\circ}
−
18
0
∘
and
18
0
∘
180^{\circ}
18
0
∘
.
\newline
θ
=
□
∘
\theta=\square^{\circ}
θ
=
□
∘
Get tutor help
A complex number
z
1
z_{1}
z
1
has a magnitude
∣
z
1
∣
=
4
\left|z_{1}\right|=4
∣
z
1
∣
=
4
and an angle
θ
1
=
33
0
∘
\theta_{1}=330^{\circ}
θ
1
=
33
0
∘
.
\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
Express
a
+
b
i
a+b i
a
+
bi
in exact terms.
\newline
z
1
=
□
z_{1}=\square
z
1
=
□
Get tutor help
z
=
−
3
−
6
i
z=-3-6 i
z
=
−
3
−
6
i
\newline
Find the angle
θ
\theta
θ
(in radians) that
z
z
z
makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express
θ
\theta
θ
between
−
π
-\pi
−
π
and
π
\pi
π
.
\newline
θ
=
\theta=
θ
=
Get tutor help
The following are all angle measures (in degrees, rounded to the nearest tenth) whose sine is
0
0
0
.
36
36
36
.
\newline
Which is the principal value of
arcsin
(
0.36
)
\arcsin (0.36)
arcsin
(
0.36
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
21.
1
∘
21.1^{\circ}
21.
1
∘
\newline
(B)
381.
1
∘
381.1^{\circ}
381.
1
∘
\newline
(C)
741.
1
∘
741.1^{\circ}
741.
1
∘
\newline
(D)
1101.
1
∘
1101.1^{\circ}
1101.
1
∘
Get tutor help
The following are all angle measures (in degrees, rounded to the nearest tenth) whose sine is
−
0
-0
−
0
.
71
71
71
.
\newline
Which is the principal value of
arcsin
(
−
0.71
)
\arcsin (-0.71)
arcsin
(
−
0.71
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
405.
2
∘
-405.2^{\circ}
−
405.
2
∘
\newline
(B)
−
45.
2
∘
-45.2^{\circ}
−
45.
2
∘
\newline
(C)
314.
8
∘
314.8^{\circ}
314.
8
∘
\newline
(D)
674.
8
∘
674.8^{\circ}
674.
8
∘
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1
2
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