If sin theta = 12/13 find the value of cos theta+5 sin theta by sin theta - cos Theta if 0 < theta < 90°
Answers
Step-by-step explanation:
can be in the first quadrant
0
≤
θ
≤
90
or the fourth quadrant
270
≤
θ
≤
360
If
θ
is in the first quadrant,
then
sin
θ
=
5
13
cos
θ
=
12
13
tan
θ
=
5
12
Therefore,
sin
2
θ
=
2
sin
θ
cos
θ
=
2
×
5
13
×
12
13
=
120
169
cos
2
θ
=
cos
2
θ
−
sin
2
θ
=
(
12
13
)
2
−
(
5
13
)
2
=
144
169
−
25
169
=
119
169
If
θ
is in the fourth quadrant,
then
sin
θ
=
−
5
13
cos
θ
=
12
13
tan
θ
=
−
5
12
Therefore,
sin
2
θ
=
2
sin
θ
cos
θ
=
2
×
−
5
13
×
12
13
=
−
120
169
cos
2
θ
=
cos
2
θ
−
sin
2
θ
=
(
12
13
)
2
−
(
−
5
13
)
2
=
144
169
−
25
169
=
119
169
Answer:
Given,
sinA=
13
12
We know that,
sinA=
Hypotenuse
oppositeSide
From Pythagoras theorem,
(Hypotenuse)
2
=(oppositeSide)
2
+(adjacentSide)
2
13
2
=12
2
+(adjacentSide)
2
(adjacentSide)
2
=169−144=25
(adjacentSide)=5
cosA=
Hypotenuse
AdjacentSide
=
13
5
tanA=
AdjacentSide
OppositeSide
=
5
12
Therefore,
2sinθcosθ
sin
2
θ−cos
2
θ
×
tan
2
θ
1
=
2(
13
12
)(
13
5
)
(
13
12
)
2
−(
13
5
)
2
×
(
5
12
)
2
1
=
2(
13
12
)(
13
5
)
(
169
144
)−(
169
25
)
×
144
25
=
(
169
120
)
(
169
144−25
)
×
144
25
=
120
119
×
144
25
=
24
119
×
144
5
=
3456
595
Step-by-step explanation:
i hope it's helpful for you