Answer:
We are given that s
i
n
(
x
)
=
1
2
. We can solve this equation for x by taking the sine inverse of both sides of the equation.
s
i
n
−
1
(
s
i
n
(
x
)
)
=
s
i
n
−
1
(
1
2
)
Simplifying the left-hand side gives the following:
x
=
s
i
n
−
1
(
1
2
)
Thus, x is the angle, such that
s
i
n
(
x
)
=
1
2
.
In trigonometry, there are special angles that have well known trigonometric values, and one of these angles is 30°. For an angle of measure 30°, we have the following:
s
i
n
(
30
∘
)
=
1
2
c
o
s
(
30
∘
)
=
√
3
2
Since
s
i
n
(
30
∘
)
=
1
2
, we have that
s
i
n
−
1
(
1
2
)
=
30
∘
, so x = 30°.
As we just saw, c
o
s
(
30
∘
)
=
√
3
2
. To find tan(30°), we will use the trigonometric identity that
t
a
n
θ
=s
i
n
θ
c
o
s
θ
. Thus, we have the following:
t
a
n(
30
∘
)
=
s
i
n
(
30
∘
)
c
o
s
(
30
∘
)
=
1
2
√
3
2
=
1
2
⋅
2
√
3
=
1
√
3
We get that
t
a
n
(
30
∘
)
=
1
√
3
. Thus, all together, we have that if s
i
n
(
x
)
=
1
2
, then
c
o
s
(
x
)
=
√
3
2 and t
a
n
(
x
)
=
1
√
3
.We are given that s
i
n
(
x
)
=
1
2
. We can solve this equation for x by taking the sine inverse of both sides of the equation.
s
i
n
−
1
(
s
i
n
(
x
)
)
=
s
i
n
−
1
(
1
2
)
Simplifying the left-hand side gives the following:
x
=
s
i
n
−
1
(
1
2
)
Thus, x is the angle, such that s
i
n
(
x
)
=
1
2
.
In trigonometry, there are special angles that have well known trigonometric values, and one of these angles is 30°. For an angle of measure 30°, we have the following:
s
i
n
(
30
∘
)
=1
2
c
o
s
(
30
∘
)
=
√
3
2 Since s
i
n
(
30
∘
)
=
1
2
, we have that s
i
n
−
1
(
1
2
)
=
30
∘
, so x = 30°.
As we just saw, c
o
s
(
30
∘
)
=
√
3
2
. To find tan(30°), we will use the trigonometric identity that t
a
n
θ
=
s
i
n
θ
c
o
s
θ
. Thus, we have the following:
t
a
n
(
30
∘
)
=
s
i
n
(
30
∘
)
c
o
s
(
30
∘
)
=
1
2
√
3
2
=
1
2
⋅
2
√
3
=
1
√
3
Step-by-step explanation:
n trigonometry, the inverse sine function, denoted as
s
i
n
−
1
x
, is defined as the function that undoes the sine function. That is, s
i
n
−
1
x is equal to the the angle, θ, such that s
in
θ
=
x
, and s
i
n
−
1
(
s
i
n
(
θ
))
=
θ
. We can use this definition to determine the angle that corresponds to a specific sine value
