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Math Problems
Algebra 2
Find trigonometric ratios using multiple identities
If
tan
(
x
)
=
5
6
\tan (x)=\frac{5}{6}
tan
(
x
)
=
6
5
(in Quadrant-I), find
\newline
cos
(
2
x
)
=
\cos (2 x)=
cos
(
2
x
)
=
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Question
\newline
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\newline
Show Examples
\newline
Given that
cos
A
=
2
3
\cos A=\frac{\sqrt{2}}{3}
cos
A
=
3
2
and
sin
B
=
3
4
\sin B=\frac{\sqrt{3}}{4}
sin
B
=
4
3
, 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.
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Let
h
h
h
be a differentiable function such that
h
(
−
5
)
=
10
h(-5)=10
h
(
−
5
)
=
10
and
h
′
(
x
)
=
2
−
e
x
+
2
x
2
h^{\prime}(x)=2-\sqrt{e^{x}+2 x^{2}}
h
′
(
x
)
=
2
−
e
x
+
2
x
2
.
\newline
What is the value of
h
(
1
)
h(1)
h
(
1
)
? Use a graphing calculator and round your answer to three decimal places.
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Given
f
=
3
f=3
f
=
3
and
g
=
10
g=10
g
=
10
, find the measure of the indicated angle.
\newline
m
∠
ϕ
=
m \angle \phi=
m
∠
ϕ
=
\newline
(Round your answer to the nearest one decimal place. Do NOT put ad
\newline
Type your answer...
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The questions below are posed in order to help you think about how to find the number of degrees in
4
π
3
\frac{4 \pi}{3}
3
4
π
radians.
\newline
What fraction of a semicircle is an angle that measures
4
π
3
\frac{4 \pi}{3}
3
4
π
radians? Express your answer as a fraction in simplest terms.
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The questions below are posed in order to help you think about how to find the number of degrees in
19
π
18
\frac{19 \pi}{18}
18
19
π
radians.
\newline
What fraction of a semicircle is an angle that measures
19
π
18
\frac{19 \pi}{18}
18
19
π
radians? Express your answer as a fraction in simplest terms.
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u
⃗
=
(
7
,
−
4
)
\vec{u}=(7,-4)
u
=
(
7
,
−
4
)
\newline
Find the direction angle of
u
⃗
\vec{u}
u
.
\newline
Enter your answer as an angle in degrees between
0
∘
0^{\circ}
0
∘
and
36
0
∘
360^{\circ}
36
0
∘
rounded to the nearest hundredth.
\newline
θ
=
□
°
\theta= \square \degree
θ
=
□
°
\newline
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Given that
tan
x
=
2
5
\tan x=\frac{2}{\sqrt{5}}
tan
x
=
5
2
and
sin
y
=
4
5
\sin y=\frac{4}{5}
sin
y
=
5
4
, 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 that
sin
A
=
7
6
\sin A=\frac{\sqrt{7}}{6}
sin
A
=
6
7
and
tan
B
=
3
\tan B=\sqrt{3}
tan
B
=
3
, 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
tan
x
=
6
\tan x=6
tan
x
=
6
and
cos
y
=
3
13
\cos y=\frac{3}{\sqrt{13}}
cos
y
=
13
3
, 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 that
sin
A
=
6
61
\sin A=\frac{6}{\sqrt{61}}
sin
A
=
61
6
and
cos
B
=
1
2
\cos B=\frac{1}{\sqrt{2}}
cos
B
=
2
1
, 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
tan
x
=
1
3
\tan x=\frac{1}{\sqrt{3}}
tan
x
=
3
1
and
cos
y
=
5
5
\cos y=\frac{\sqrt{5}}{5}
cos
y
=
5
5
, 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 that
cos
x
=
5
5
\cos x=\frac{\sqrt{5}}{5}
cos
x
=
5
5
and
sin
y
=
1
3
\sin y=\frac{1}{3}
sin
y
=
3
1
, 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
sin
A
=
22
5
\sin A=\frac{\sqrt{22}}{5}
sin
A
=
5
22
and
sin
B
=
7
3
\sin B=\frac{\sqrt{7}}{3}
sin
B
=
3
7
, 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
=
4
5
\cos x=\frac{4}{5}
cos
x
=
5
4
and
sin
y
=
8
3
\sin y=\frac{\sqrt{8}}{3}
sin
y
=
3
8
, 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
tan
x
=
3
2
\tan x=\frac{3}{2}
tan
x
=
2
3
and
cos
y
=
6
37
\cos y=\frac{6}{\sqrt{37}}
cos
y
=
37
6
, 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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If
cos
A
=
28
53
\cos A=\frac{28}{53}
cos
A
=
53
28
and
tan
B
=
5
12
\tan B=\frac{5}{12}
tan
B
=
12
5
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
=
21
29
\cos A=\frac{21}{29}
cos
A
=
29
21
and
tan
B
=
5
12
\tan B=\frac{5}{12}
tan
B
=
12
5
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
=
20
29
\sin A=\frac{20}{29}
sin
A
=
29
20
and
cos
B
=
28
53
\cos B=\frac{28}{53}
cos
B
=
53
28
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
=
24
7
\tan A=\frac{24}{7}
tan
A
=
7
24
and
cos
B
=
12
37
\cos B=\frac{12}{37}
cos
B
=
37
12
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
=
35
12
\tan A=\frac{35}{12}
tan
A
=
12
35
and
cos
B
=
5
13
\cos B=\frac{5}{13}
cos
B
=
13
5
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
=
35
37
\sin A=\frac{35}{37}
sin
A
=
37
35
and
cos
B
=
8
17
\cos B=\frac{8}{17}
cos
B
=
17
8
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
=
9
41
\sin A=\frac{9}{41}
sin
A
=
41
9
and
cos
B
=
21
29
\cos B=\frac{21}{29}
cos
B
=
29
21
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
=
4
3
\tan A=\frac{4}{3}
tan
A
=
3
4
and
cos
B
=
8
17
\cos B=\frac{8}{17}
cos
B
=
17
8
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
=
9
41
\cos A=\frac{9}{41}
cos
A
=
41
9
and
tan
B
=
5
12
\tan B=\frac{5}{12}
tan
B
=
12
5
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
sin
x
=
3
58
\sin x=\frac{3}{\sqrt{58}}
sin
x
=
58
3
and
tan
y
=
1
\tan y=1
tan
y
=
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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Given that
sin
x
=
1
26
\sin x=\frac{1}{\sqrt{26}}
sin
x
=
26
1
and
sin
y
=
1
5
\sin y=\frac{1}{\sqrt{5}}
sin
y
=
5
1
, 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
tan
A
=
2
\tan A=2
tan
A
=
2
and
sin
B
=
5
41
\sin B=\frac{5}{\sqrt{41}}
sin
B
=
41
5
, 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
sin
x
=
1
2
\sin x=\frac{1}{\sqrt{2}}
sin
x
=
2
1
and
cos
y
=
3
10
\cos y=\frac{3}{\sqrt{10}}
cos
y
=
10
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
tan
x
=
1
3
\tan x=\frac{1}{\sqrt{3}}
tan
x
=
3
1
and
sin
y
=
28
6
\sin y=\frac{\sqrt{28}}{6}
sin
y
=
6
28
, 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 that
cos
A
=
23
5
\cos A=\frac{\sqrt{23}}{5}
cos
A
=
5
23
and
sin
B
=
21
6
\sin B=\frac{\sqrt{21}}{6}
sin
B
=
6
21
, 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
tan
y
=
3
7
\tan y=\frac{3}{\sqrt{7}}
tan
y
=
7
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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If
cos
A
=
35
37
\cos A=\frac{35}{37}
cos
A
=
37
35
and
sin
B
=
28
53
\sin B=\frac{28}{53}
sin
B
=
53
28
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
=
40
9
\tan A=\frac{40}{9}
tan
A
=
9
40
and
sin
B
=
15
17
\sin B=\frac{15}{17}
sin
B
=
17
15
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
=
12
35
\tan A=\frac{12}{35}
tan
A
=
35
12
and
cos
B
=
15
17
\cos B=\frac{15}{17}
cos
B
=
17
15
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
=
11
61
\sin A=\frac{11}{61}
sin
A
=
61
11
and
cos
B
=
20
29
\cos B=\frac{20}{29}
cos
B
=
29
20
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
=
21
29
\cos A=\frac{21}{29}
cos
A
=
29
21
and
tan
B
=
9
40
\tan B=\frac{9}{40}
tan
B
=
40
9
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
=
40
41
\sin A=\frac{40}{41}
sin
A
=
41
40
and
cos
B
=
20
29
\cos B=\frac{20}{29}
cos
B
=
29
20
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
=
7
25
\cos A=\frac{7}{25}
cos
A
=
25
7
and
tan
B
=
3
4
\tan B=\frac{3}{4}
tan
B
=
4
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
tan
A
=
35
12
\tan A=\frac{35}{12}
tan
A
=
12
35
and
cos
B
=
21
29
\cos B=\frac{21}{29}
cos
B
=
29
21
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
=
12
5
\tan A=\frac{12}{5}
tan
A
=
5
12
and
cos
B
=
8
17
\cos B=\frac{8}{17}
cos
B
=
17
8
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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Point
F
\mathrm{F}
F
is located at
−
11
-11
−
11
. Point
G
\mathrm{G}
G
is
9
9
9
less than Point
F
\mathrm{F}
F
. Where is
G
\mathrm{G}
G
located?
\newline
G
=
□
\mathrm{G}= \square
G
=
□
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The angle
θ
1
\theta_{1}
θ
1
is located in Quadrant 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.
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\newline
Given
cot
A
=
−
5
171
\cot A=-\frac{5}{\sqrt{171}}
cot
A
=
−
171
5
and that angle
A
A
A
is in Quadrant IV, find the exact value of
sin
A
\sin A
sin
A
n simplest radical form using a rational denominator.
\newline
Answer____
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If
8
x
−
y
=
−
10
\mathbf{8 x}-\mathbf{y}=-\mathbf{1 0}
8x
−
y
=
−
10
is a true equation, what would be the value of
4
+
8
x
−
y
?
\mathbf{4}+\mathbf{8 x}-\mathbf{y} ?
4
+
8x
−
y
?
\newline
Answer:
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If
9
x
−
10
y
=
7
\mathbf{9 x}-\mathbf{1 0 y}=\mathbf{7}
9x
−
10y
=
7
and
−
x
−
9
y
=
6
-\mathbf{x}-\mathbf{9 y}=\mathbf{6}
−
x
−
9y
=
6
are true equations, what would be the value of
−
10
x
+
y
-\mathbf{1 0 x}+\mathbf{y}
−
10x
+
y
?
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Answer:
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If
2
x
+
10
y
=
3
\mathbf{2 x}+\mathbf{1 0 y}=\mathbf{3}
2x
+
10y
=
3
and
−
5
x
−
8
y
=
7
-\mathbf{5 x}-\mathbf{8 y}=\mathbf{7}
−
5x
−
8y
=
7
are true equations, what would be the value of
7
x
+
18
y
\mathbf{7 x}+\mathbf{1 8 y}
7x
+
18y
?
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For the following equation, evaluate
d
y
d
x
\frac{d y}{d x}
d
x
d
y
when
x
=
−
3
x=-3
x
=
−
3
.
\newline
y
=
3
x
2
+
3
y=3 x^{2}+3
y
=
3
x
2
+
3
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Answer:
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The derivative of the function
f
f
f
is defined by
f
′
(
x
)
=
(
x
3
+
3
x
)
cos
(
3
x
+
4
)
f^{\prime}(x)=\left(x^{3}+3 x\right) \cos (3 x+4)
f
′
(
x
)
=
(
x
3
+
3
x
)
cos
(
3
x
+
4
)
. If
f
(
3
)
=
−
4
f(3)=-4
f
(
3
)
=
−
4
, then use a calculator to find the value of
f
(
−
2
)
f(-2)
f
(
−
2
)
to the nearest thousandth.
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The derivative of the function
f
f
f
is defined by
f
′
(
x
)
=
(
x
2
+
3
)
sin
(
3
x
)
f^{\prime}(x)=\left(x^{2}+3\right) \sin (3 x)
f
′
(
x
)
=
(
x
2
+
3
)
sin
(
3
x
)
. If
f
(
2
)
=
7
f(2)=7
f
(
2
)
=
7
, then use a calculator to find the value of
f
(
6
)
f(6)
f
(
6
)
to the nearest thousandth.
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Answer:
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