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Byju's Answer
Standard XII
Mathematics
Trigonometric Ratios of Compound Angles
If cosθ+3 sin...
Question
If
cos
θ
+
3
sin
θ
=
2
,
then
θ
=
(a)
π
/
3
(b)
2
π
/
3
(c)
4
π
/
6
(d)
5
π
/
12
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Solution
(a)
π
/
3
Given:
cos
θ
+
3
sin
θ
=
2
...(i)
This equation is of the form
a
cos
θ
+
b
sin
θ
=
c
, where
a
=
1
,
b
=
3
and
c
=
2
.
Let:
a
=
r
cos
α
and
b
=
sin
α
Now,
1
=
r
cos
α
,
3
=
r
sin
α
⇒
r
=
a
2
+
b
2
=
1
+
3
=
4
=
2
And,
tan
α
=
b
a
⇒
tan
α
=
3
1
⇒
tan
α
=
3
⇒
α
=
π
3
On putting
a
=
1
=
r
cos
α
and
b
=
3
=
r
sin
α
in equation (i), we get:
r
cos
θ
cos
α
+
r
sin
θ
sin
α
=
2
⇒
r
cos
(
θ
-
α
)
=
2
⇒
2
cos
θ
-
π
3
=
2
⇒
cos
θ
-
π
3
=
1
⇒
cos
θ
-
π
3
=
cos
0
⇒
θ
-
π
3
=
2
n
π
±
0
⇒
θ
=
2
nπ
±
π
3
For
n
=
0
,
θ
=
π
3
.
∴
θ
=
π
3
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0
Similar questions
Q.
If
cos
x
+
3
sin
x
=
2
,
then
x
=
(a)
π
/
3
(b)
2
π
/
3
(c)
4
π
/
6
(d)
5
π
/
12
Q.
The smallest value of θ satisfying the equation
3
cot
θ
+
tan
θ
=
4
is
(a)
2
π
/
3
(b)
π
/
3
(c)
π
/
6
(d)
π
/
12
Q.
Let
S
=
{
θ
∈
[
−
2
π
,
2
π
]
:
2
cos
2
θ
+
3
sin
θ
=
0
}
.
Then the sum of the elements of
S
is:
Q.
If
−
1
+
√
−
3
=
r
e
θ
, then θ is equal to
Q.
If -1+
√
−
3
=r
e
i
θ
, then
θ
is equal to