Answer: -4/3
Explanation:
To determine the values of
sin
(
�
)
sin(θ),
cos
(
�
)
cos(θ), and
tan
(
�
)
tan(θ) for an angle
�
θ in standard position, we can use the coordinates of the point where the terminal side of the angle intersects the unit circle.
Let
�
(
�
,
�
)
P(x,y) be the point where the terminal side intersects the unit circle. In this case,
�
(
3
,
−
4
)
P(3,−4).
The distance from the origin to the point
(
�
,
�
)
(x,y) on the unit circle is given by the radius, which is 1 in this case.
The Pythagorean theorem tells us that:
�
=
�
2
+
�
2
r=
x
2
+y
2
For the point
(
3
,
−
4
)
(3,−4):
�
=
3
2
+
(
−
4
)
2
=
9
+
16
=
25
=
5
r=
3
2
+(−4)
2
=
9+16
=
25
=5
Now, we can use these values to find
sin
(
�
)
sin(θ),
cos
(
�
)
cos(θ), and
tan
(
�
)
tan(θ):
sin
(
�
)
=
�
�
=
−
4
5
sin(θ)=
r
y
=
5
−4
cos
(
�
)
=
�
�
=
3
5
cos(θ)=
r
x
=
5
3
tan
(
�
)
=
�
�
=
−
4
3
tan(θ)=
x
y
=
3
−4
So, for the angle
�
θ in standard position whose terminal side passes through the point
(
3
,
−
4
)
(3,−4):
sin
(
�
)
=
−
4
5
sin(θ)=
5
−4
cos
(
�
)
=
3
5
cos(θ)=
5
3
tan
(
�
)
=
−
4
3
tan(θ)=
3
−4
To determine the values of
sin
(
�
)
sin(θ),
cos
(
�
)
cos(θ), and
tan
(
�
)
tan(θ) for an angle
�
θ in standard position, we can use the coordinates of the point where the terminal side of the angle intersects the unit circle.
Let
�
(
�
,
�
)
P(x,y) be the point where the terminal side intersects the unit circle. In this case,
�
(
3
,
−
4
)
P(3,−4).
The distance from the origin to the point
(
�
,
�
)
(x,y) on the unit circle is given by the radius, which is 1 in this case.
The Pythagorean theorem tells us that:
�
=
�
2
+
�
2
r=
x
2
+y
2
For the point
(
3
,
−
4
)
(3,−4):
�
=
3
2
+
(
−
4
)
2
=
9
+
16
=
25
=
5
r=
3
2
+(−4)
2
=
9+16
=
25
=5
Now, we can use these values to find
sin
(
�
)
sin(θ),
cos
(
�
)
cos(θ), and
tan
(
�
)
tan(θ):
sin
(
�
)
=
�
�
=
−
4
5
sin(θ)=
r
y
=
5
−4
cos
(
�
)
=
�
�
=
3
5
cos(θ)=
r
x
=
5
3
tan
(
�
)
=
�
�
=
−
4
3
tan(θ)=
x
y
=
3
−4
So, for the angle
�
θ in standard position whose terminal side passes through the point
(
3
,
−
4
)
(3,−4):
sin
(
�
)
=
−
4
5
sin(θ)=
5
−4
cos
(
�
)
=
3
5
cos(θ)=
5
3
tan
(
�
)
=
−
4
3
tan(θ)=
3
−4
To determine the values of
sin
(
�
)
sin(θ),
cos
(
�
)
cos(θ), and
tan
(
�
)
tan(θ) for an angle
�
θ in standard position, we can use the coordinates of the point where the terminal side of the angle intersects the unit circle.
Let
�
(
�
,
�
)
P(x,y) be the point where the terminal side intersects the unit circle. In this case,
�
(
3
,
−
4
)
P(3,−4).
The distance from the origin to the point
(
�
,
�
)
(x,y) on the unit circle is given by the radius, which is 1 in this case.
The Pythagorean theorem tells us that:
�
=
�
2
+
�
2
r=
x
2
+y
2
For the point
(
3
,
−
4
)
(3,−4):
�
=
3
2
+
(
−
4
)
2
=
9
+
16
=
25
=
5
r=
3
2
+(−4)
2
=
9+16
=
25
=5
Now, we can use these values to find
sin
(
�
)
sin(θ),
cos
(
�
)
cos(θ), and
tan
(
�
)
tan(θ):
sin
(
�
)
=
�
�
=
−
4
5
sin(θ)=
r
y
=
5
−4
cos
(
�
)
=
�
�
=
3
5
cos(θ)=
r
x
=
5
3
tan
(
�
)
=
�
�
=
−
4
3
tan(θ)=
x
y
=
3
−4
So, for the angle
�
θ in standard position whose terminal side passes through the point
(
3
,
−
4
)
(3,−4):
sin
(
�
)
=
−
4
5
sin(θ)=
5
−4
cos
(
�
)
=
3
5
cos(θ)=
5
3
tan
(
�
)
=
−
4
3
tan(θ)=
3
−4