if sin^2 + 3 cos^2=4 then show that tanO°=1/√3
Answers
Answer:
7sin 2 x+3cos
2 x=4
4sin 2 x+3sin
2 x+3cos 2 x=4
4sin 2 x+3=4
4sin 2
x=1
sin 2 x= 41
sinx= 21
or sinx=− 21
Taking the positive root
x= 6π
tan( 6π )= 31
Hence proved.
okk.. bye
Step-by-step explanation:
We have:
sin
2
(
θ
)
+
3
cos
2
(
θ
)
=
4
which really is just:
sin
2
(
θ
)
+
cos
2
(
θ
)
+
2
cos
2
(
θ
)
=
4
and using the identity
sin
2
(
x
)
+
cos
2
(
x
)
=
1
for all
x
, we get:
1
+
2
cos
2
(
θ
)
=
4
(eq.A)
We carry on and simplify eq.A until we get an expression for
cos
(
θ
)
:
2
cos
2
(
θ
)
=
3
cos
2
(
θ
)
=
3
2
cos
(
θ
)
=
±
√
3
2
We also notice that
cos
2
(
θ
)
=
1
−
sin
2
(
θ
)
, so (eq.A) becomes:
1
+
2
cos
2
(
θ
)
=
1
+
2
(
1
−
sin
2
(
θ
)
)
=
4
i.e.
3
−
2
sin
2
(
θ
)
=
4
i.e.
sin
2
(
θ
)
=
−
1
2
sin
(
θ
)
=
±
√
1
2
(it should really be 'minus-plus' instead but the symbol does not exist here in Socratic).
So, now it is simply a matter of plugging these in the
tan
function:
tan
(
θ
)
=
sin
(
θ
)
cos
(
θ
)
=
±
√
1
2
±
√
3
2
=
±
(
1
√
3
)
=
±
√
3
3
(it is more conventional to write it like this without the square-root in the denominator).