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Math Problems
Precalculus
Find trigonometric ratios using multiple identities
Write the following in terms of
sin
(
x
)
\sin (x)
sin
(
x
)
and
cos
(
x
)
\cos (x)
cos
(
x
)
, and then simplify if possible. Leave your answer in terms of sines and cosines only.
\newline
csc
(
x
)
cot
(
x
)
=
\csc (x) \cot (x)=
csc
(
x
)
cot
(
x
)
=
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For the rotation
−
55
5
∘
-555^{\circ}
−
55
5
∘
, 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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If
tan
A
=
60
11
\tan A=\frac{60}{11}
tan
A
=
11
60
and
sin
B
=
3
5
\sin B=\frac{3}{5}
sin
B
=
5
3
and angles A and B are in Quadrant I, find the value of
tan
(
A
+
B
)
\tan (A+B)
tan
(
A
+
B
)
.
\newline
Answer:
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If
sin
A
=
12
37
\sin A=\frac{12}{37}
sin
A
=
37
12
and
tan
B
=
7
24
\tan B=\frac{7}{24}
tan
B
=
24
7
and angles A and B are in Quadrant I, find the value of
tan
(
A
−
B
)
\tan (A-B)
tan
(
A
−
B
)
.
\newline
Answer:
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If
cos
A
=
12
37
\cos A=\frac{12}{37}
cos
A
=
37
12
and
sin
B
=
3
5
\sin B=\frac{3}{5}
sin
B
=
5
3
and angles A and B are in Quadrant I, find the value of
tan
(
A
−
B
)
\tan (A-B)
tan
(
A
−
B
)
.
\newline
Answer:
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Given that
cos
x
=
5
4
\cos x=\frac{\sqrt{5}}{4}
cos
x
=
4
5
and
sin
y
=
1
4
\sin y=\frac{1}{4}
sin
y
=
4
1
, 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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If
cos
A
=
4
5
\cos A=\frac{4}{5}
cos
A
=
5
4
and
tan
B
=
11
60
\tan B=\frac{11}{60}
tan
B
=
60
11
and angles A and B are in Quadrant I, find the value of
tan
(
A
+
B
)
\tan (A+B)
tan
(
A
+
B
)
.
\newline
Answer:
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If
tan
A
=
28
45
\tan A=\frac{28}{45}
tan
A
=
45
28
and
cos
B
=
4
5
\cos B=\frac{4}{5}
cos
B
=
5
4
and angles A and B are in Quadrant I, find the value of
tan
(
A
−
B
)
\tan (A-B)
tan
(
A
−
B
)
.
\newline
Answer:
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Find the argument of the complex number
−
3
3
+
9
i
-3 \sqrt{3}+9 i
−
3
3
+
9
i
in the interval
0
≤
θ
<
2
π
0 \leq \theta<2 \pi
0
≤
θ
<
2
π
. Express your answer in terms of
π
\pi
π
.
\newline
Answer:
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Find the argument of the complex number
3
+
3
i
3+\sqrt{3} i
3
+
3
i
in the interval
0
≤
θ
<
2
π
0 \leq \theta<2 \pi
0
≤
θ
<
2
π
.
\newline
Express your answer in terms of
π
\pi
π
.
\newline
Answer:
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Find the argument of the complex number
3
−
3
i
3-\sqrt{3} i
3
−
3
i
in the interval
0
≤
θ
<
2
π
0 \leq \theta<2 \pi
0
≤
θ
<
2
π
.
\newline
Express your answer in terms of
π
\pi
π
.
\newline
Answer:
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The area of a triangle is
866
866
866
. Two of the side lengths are
33
33
33
and
98
98
98
and the included angle is acute. Find the measure of the included angle, to the nearest tenth of a degree.
\newline
Answer:
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Given
tan
A
=
−
11
60
\tan A=-\frac{11}{60}
tan
A
=
−
60
11
and that angle
A
A
A
is in Quadrant IV, find the exact value of
csc
A
\csc A
csc
A
in simplest radical form using a rational denominator.
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If
a
1
=
0
,
a
2
=
2
a_{1}=0, a_{2}=2
a
1
=
0
,
a
2
=
2
and
a
n
=
a
n
−
1
+
3
a
n
−
2
a_{n}=a_{n-1}+3 a_{n-2}
a
n
=
a
n
−
1
+
3
a
n
−
2
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
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Given that
f
(
x
)
=
2
x
,
g
(
x
)
=
x
−
2
f(x)=2 x, g(x)=x-2
f
(
x
)
=
2
x
,
g
(
x
)
=
x
−
2
and
h
(
x
)
=
−
3
f
(
x
+
3
)
−
g
(
x
)
h(x)=-3 f(x+3)-g(x)
h
(
x
)
=
−
3
f
(
x
+
3
)
−
g
(
x
)
, then what is the value of
h
(
1
)
h(1)
h
(
1
)
?
\newline
Answer:
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If
f
(
1
)
=
9
f(1)=9
f
(
1
)
=
9
and
f
(
n
)
=
n
f
(
n
−
1
)
−
2
f(n)=n f(n-1)-2
f
(
n
)
=
n
f
(
n
−
1
)
−
2
then find the value of
f
(
5
)
f(5)
f
(
5
)
.
\newline
Answer:
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Given that
f
(
x
)
=
−
2
x
,
g
(
x
)
=
x
−
2
f(x)=-2 x, g(x)=x-2
f
(
x
)
=
−
2
x
,
g
(
x
)
=
x
−
2
and
h
(
x
)
=
−
2
f
(
x
)
+
3
g
(
x
+
1
)
h(x)=-2 f(x)+3 g(x+1)
h
(
x
)
=
−
2
f
(
x
)
+
3
g
(
x
+
1
)
, then what is the value of
h
(
6
)
h(6)
h
(
6
)
?
\newline
Answer:
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Given that
f
(
x
)
=
x
+
1
,
g
(
x
)
=
−
3
x
f(x)=x+1, g(x)=-3 x
f
(
x
)
=
x
+
1
,
g
(
x
)
=
−
3
x
and
h
(
x
)
=
−
f
(
x
)
+
3
g
(
x
+
3
)
h(x)=-f(x)+3 g(x+3)
h
(
x
)
=
−
f
(
x
)
+
3
g
(
x
+
3
)
, then what is the value of
h
(
3
)
h(3)
h
(
3
)
?
\newline
Answer:
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Find an angle
θ
\theta
θ
coterminal to
93
8
∘
938^{\circ}
93
8
∘
, where
0
∘
≤
θ
<
36
0
∘
0^{\circ} \leq \theta<360^{\circ}
0
∘
≤
θ
<
36
0
∘
.
\newline
Answer:
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Given
f
(
x
)
=
−
4
x
−
1
f(x)=-4 x-1
f
(
x
)
=
−
4
x
−
1
, find
f
(
−
5
)
f(-5)
f
(
−
5
)
.
\newline
Answer:
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Given
g
(
x
)
=
−
x
−
2
g(x)=-x-2
g
(
x
)
=
−
x
−
2
, find
g
(
−
6
)
g(-6)
g
(
−
6
)
.
\newline
Answer:
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Given
g
(
x
)
=
−
x
+
2
g(x)=-x+2
g
(
x
)
=
−
x
+
2
, find
g
(
−
6
)
g(-6)
g
(
−
6
)
.
\newline
Answer:
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Given
h
(
x
)
=
−
x
−
4
h(x)=-x-4
h
(
x
)
=
−
x
−
4
, find
h
(
−
2
)
h(-2)
h
(
−
2
)
.
\newline
Answer:
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Given
h
(
x
)
=
−
x
+
5
h(x)=-x+5
h
(
x
)
=
−
x
+
5
, find
h
(
−
2
)
h(-2)
h
(
−
2
)
.
\newline
Answer:
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Given
h
(
x
)
=
−
4
x
−
3
h(x)=-4 x-3
h
(
x
)
=
−
4
x
−
3
, find
h
(
−
3
)
h(-3)
h
(
−
3
)
.
\newline
Answer:
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Given
g
(
x
)
=
−
3
x
−
2
g(x)=-3 x-2
g
(
x
)
=
−
3
x
−
2
, find
g
(
−
2
)
g(-2)
g
(
−
2
)
.
\newline
Answer:
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Evaluate the left hand side to find the value of
a
a
a
in the equation in simplest form.
\newline
x
3
4
x
3
=
x
a
x^{\frac{3}{4}} x^{3}=x^{a}
x
4
3
x
3
=
x
a
\newline
Answer:
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Evaluate the left hand side to find the value of
a
a
a
in the equation in simplest form.
\newline
x
1
2
x
6
=
x
a
x^{\frac{1}{2}} x^{6}=x^{a}
x
2
1
x
6
=
x
a
\newline
Answer:
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If
f
(
1
)
=
7
f(1)=7
f
(
1
)
=
7
and
f
(
n
)
=
2
f
(
n
−
1
)
−
2
f(n)=2 f(n-1)-2
f
(
n
)
=
2
f
(
n
−
1
)
−
2
then find the value of
f
(
3
)
f(3)
f
(
3
)
.
\newline
Answer:
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If
a
1
=
9
a_{1}=9
a
1
=
9
and
a
n
=
−
4
a
n
−
1
+
4
a_{n}=-4 a_{n-1}+4
a
n
=
−
4
a
n
−
1
+
4
then find the value of
a
3
a_{3}
a
3
.
\newline
Answer:
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If
a
1
=
7
a_{1}=7
a
1
=
7
and
a
n
=
−
2
a
n
−
1
−
2
a_{n}=-2 a_{n-1}-2
a
n
=
−
2
a
n
−
1
−
2
then find the value of
a
3
a_{3}
a
3
.
\newline
Answer:
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If
a
1
=
1
a_{1}=1
a
1
=
1
and
a
n
=
3
a
n
−
1
+
2
a_{n}=3 a_{n-1}+2
a
n
=
3
a
n
−
1
+
2
then find the value of
a
5
a_{5}
a
5
.
\newline
Answer:
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If
a
1
=
10
a_{1}=10
a
1
=
10
and
a
n
=
3
a
n
−
1
−
5
a_{n}=3 a_{n-1}-5
a
n
=
3
a
n
−
1
−
5
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
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If
a
1
=
8
a_{1}=8
a
1
=
8
and
a
n
=
−
4
a
n
−
1
−
n
a_{n}=-4 a_{n-1}-n
a
n
=
−
4
a
n
−
1
−
n
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
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If
a
1
=
1
a_{1}=1
a
1
=
1
and
a
n
=
3
a
n
−
1
−
n
a_{n}=3 a_{n-1}-n
a
n
=
3
a
n
−
1
−
n
then find the value of
a
5
a_{5}
a
5
.
\newline
Answer:
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If
a
1
=
9
a_{1}=9
a
1
=
9
and
a
n
+
1
=
4
a
n
+
4
a_{n+1}=4 a_{n}+4
a
n
+
1
=
4
a
n
+
4
then find the value of
a
3
a_{3}
a
3
.
\newline
Answer:
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If
a
1
=
7
a_{1}=7
a
1
=
7
and
a
n
+
1
=
−
3
a
n
+
5
a_{n+1}=-3 a_{n}+5
a
n
+
1
=
−
3
a
n
+
5
then find the value of
a
3
a_{3}
a
3
.
\newline
Answer:
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If
a
1
=
10
a_{1}=10
a
1
=
10
and
a
n
+
1
=
−
3
a
n
+
5
a_{n+1}=-3 a_{n}+5
a
n
+
1
=
−
3
a
n
+
5
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
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If
a
1
=
3
a_{1}=3
a
1
=
3
and
a
n
+
1
=
3
a
n
+
5
a_{n+1}=3 a_{n}+5
a
n
+
1
=
3
a
n
+
5
then find the value of
a
4
a_{4}
a
4
.
\newline
Answer:
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If
a
1
=
4
a_{1}=4
a
1
=
4
and
a
n
+
1
=
2
a
n
+
4
a_{n+1}=2 a_{n}+4
a
n
+
1
=
2
a
n
+
4
then find the value of
a
3
a_{3}
a
3
.
\newline
Answer:
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Determine the value of
y
y
y
, if
x
x
x
is
25
25
25
.
\newline
y
=
x
+
11
y=\sqrt{x+11}
y
=
x
+
11
\newline
Answer:
y
=
y=
y
=
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Given the function
y
=
−
4
x
4
5
5
y=-\frac{4 \sqrt[5]{x^{4}}}{5}
y
=
−
5
4
5
x
4
, find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
. Express your answer in radical form without using negative exponents, simplifying all fractions.
\newline
Answer:
d
y
d
x
=
\frac{d y}{d x}=
d
x
d
y
=
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Given the function
y
=
−
2
3
x
4
5
y=-\frac{2}{3 \sqrt[5]{x^{4}}}
y
=
−
3
5
x
4
2
, find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
. Express your answer in radical form without using negative exponents, simplifying all fractions.
\newline
Answer:
d
y
d
x
=
\frac{d y}{d x}=
d
x
d
y
=
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Given the function
y
=
4
5
x
3
5
y=\frac{4}{5 \sqrt[5]{x^{3}}}
y
=
5
5
x
3
4
, find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
. Express your answer in radical form without using negative exponents, simplifying all fractions.
\newline
Answer:
d
y
d
x
=
\frac{d y}{d x}=
d
x
d
y
=
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Given the function
f
(
x
)
=
3
x
2
−
3
x
f(x)=\frac{3 \sqrt{x}}{2}-\frac{3}{\sqrt{x}}
f
(
x
)
=
2
3
x
−
x
3
, find
f
′
(
6
)
f^{\prime}(6)
f
′
(
6
)
. Express your answer as a single fraction in simplest radical form.
\newline
Answer:
f
′
(
6
)
=
f^{\prime}(6)=
f
′
(
6
)
=
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Given the function
f
(
x
)
=
3
2
x
−
3
x
2
f(x)=\frac{3}{2 \sqrt{x}}-\frac{3 \sqrt{x}}{2}
f
(
x
)
=
2
x
3
−
2
3
x
, find
f
′
(
3
)
f^{\prime}(3)
f
′
(
3
)
. Express your answer as a single fraction in simplest radical form.
\newline
Answer:
f
′
(
3
)
=
f^{\prime}(3)=
f
′
(
3
)
=
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Given the function
f
(
x
)
=
−
x
3
+
3
2
x
f(x)=-\sqrt{x^{3}}+\frac{3}{2 \sqrt{x}}
f
(
x
)
=
−
x
3
+
2
x
3
, find
f
′
(
3
)
f^{\prime}(3)
f
′
(
3
)
. Express your answer as a single fraction in simplest radical form.
\newline
Answer:
f
′
(
3
)
=
f^{\prime}(3)=
f
′
(
3
)
=
Get tutor help
In
△
A
B
C
,
A
C
‾
\triangle \mathrm{ABC}, \overline{A C}
△
ABC
,
A
C
is extended through point
C
\mathrm{C}
C
to point
D
,
m
∠
C
A
B
=
(
x
+
14
)
∘
\mathrm{D}, \mathrm{m} \angle C A B=(x+14)^{\circ}
D
,
m
∠
C
A
B
=
(
x
+
14
)
∘
,
m
∠
A
B
C
=
(
x
−
3
)
∘
\mathrm{m} \angle A B C=(x-3)^{\circ}
m
∠
A
BC
=
(
x
−
3
)
∘
, and
m
∠
B
C
D
=
(
4
x
−
11
)
∘
\mathrm{m} \angle B C D=(4 x-11)^{\circ}
m
∠
BC
D
=
(
4
x
−
11
)
∘
. What is the value of
x
?
x ?
x
?
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Answer:
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If
6
x
+
9
y
=
−
9
6 x+9 y=-9
6
x
+
9
y
=
−
9
is a true equation, what would be the value of
−
3
+
6
x
+
9
y
?
-3+6 x+9 y ?
−
3
+
6
x
+
9
y
?
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Answer:
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If
x
+
9
y
=
−
8
\mathbf{x}+\mathbf{9 y}=-\mathbf{8}
x
+
9y
=
−
8
is a true equation, what would be the value of
2
+
x
+
9
y
\mathbf{2}+\mathbf{x}+\mathbf{9 y}
2
+
x
+
9y
?
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Answer:
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