Question

sin teta-cos teta ÷sin teta +cos teta =3 ;sin teta 4+cos teta 4=¿

Answer 1

Answer:
sin
θ
=
3
5

Explanation:
We have :

⇒
3
sin
θ
+
4
cos
θ
=
5

Subtract
3
sin
θ
from both sides
→

⇒
4
cos
θ
=
5
−
3
sin
θ

Square both sides
→

⇒
16
cos
2
θ
=
25
+
9
sin
2
θ
−
2
(
5
)
(
3
sin
θ
)

Now substitute
cos
2
θ
=
1
−
sin
2
θ
at L.H.S
→

⇒
16
(
1
−
sin
2
θ
)
=
25
+
9
sin
2
θ
−
30
sin
θ

Rearrange the terms to get a quadratic in
sin
θ
→

⇒
9
sin
2
θ
+
16
sin
2
θ
−
30
sin
θ
+
25
−
16
=
0

⇒
25
sin
2
θ
−
30
sin
θ
+
9
=
0

Now if you look , the above quadratic equation is a perfect square using the identity
a
2
+
b
2
−
2
a
b
=
(
a
−
b
)
2
we write it as
→

⇒
(
5
sin
θ
)
2

a
2

−
2
⋅
(
5
sin
θ
)
⋅
(
3
)

−
2
a
b

+
(
3
)
2

b
2

=
0

⇒
(
5
sin
θ
−
3
)
2
=
0

⇒
5
sin
θ
−
3
=
0

⇒
sin
θ
=
3
5
.