Find the value of theta if cos theta by 1 minus sin theta + cos theta by 1 + sin theta is equals to 4 theta greater than 90°
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Answer:
θ
=
π
3
or
60
∘
Explanation:
Okay. We've got:
cos
θ
1
−
sin
θ
+
cos
θ
1
+
sin
θ
=
4
Let's ignore the
R
H
S
for now.
cos
θ
1
−
sin
θ
+
cos
θ
1
+
sin
θ
cos
θ
(
1
+
sin
θ
)
+
cos
θ
(
1
−
sin
θ
)
(
1
−
sin
θ
)
(
1
+
sin
θ
)
cos
θ
(
(
1
−
sin
θ
)
+
(
1
+
sin
θ
)
)
1
−
sin
2
θ
cos
θ
(
1
−
sin
θ
+
1
+
sin
θ
)
1
−
sin
2
θ
2
cos
θ
1
−
sin
2
θ
According to the Pythagorean Identity,
sin
2
θ
+
cos
2
θ
=
1
. So:
cos
2
θ
=
1
−
sin
2
θ
Now that we know that, we can write:
2
cos
θ
cos
2
θ
2
cos
θ
=
4
cos
θ
2
=
1
4
cos
θ
=
1
2
θ
=
cos
−
1
(
1
2
)
θ
=
π
3
, when
0
≤
θ
≤
π
.
In degrees,
θ
=
60
∘
when
0
∘
≤
θ
≤
180
∘
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