Prove that
sin²120°+cos²150°+tan²120°+cos180°-tan135°=9/2
Answers
Answer:
Step-by-step explanation:
sin
2
(
120
∘
)
+
cos
2
(
150
∘
)
+
tan
2
(
120
∘
)
+
cos
(
180
∘
)
−
tan
(
135
∘
)
=
(
√
3
2
)
2
+
(
−
√
3
2
)
2
+
(
−
√
3
)
2
+
(
−
1
)
−
(
−
1
)
=
3
4
+
3
4
+
3
−
1
+
1
=
3
2
+
3
=
4.5
enter image source here
Above, you see a unit circle (circle with radius of one). By definition,
sin
θ
=
Opposite
Hypotenuse
=
y
r
=
y
1
=
y
cos
θ
=
Adjacent
Hypotenuse
=
x
r
=
x
1
=
x
As angle
θ
varies, point
A
on the circle moves on the perimeter of the circle. Its coordinates, regardless of where on the circle it may be, can always be expressed as:
A
(
cos
θ
,
sin
θ
)
There are certain commonly used angles in trigonometry, such as
30
∘
,
45
∘
,
60
∘
,
90
∘
,
120
∘
,
e
t
c
.
that you need to memorize what their coordinates are on the unit circle.
The above angles in radian are
π
6
,
π
4
,
π
3
,
π
2
,
2
π
3
,
e
t
c
.
.
These coordinates are the
x
and
y
of the point and, as described above, are
cos
θ
and
sin
θ
.
Knowing these values in trigonometry are like knowing the multiplication table in arithmetic.
The best way to accomplish this is to have a printout of a unit circle with these measurements both in degrees and radians in front of you; and refer to them when solving trigonometric problems.
After some time, they will stay in your mind. But it is essential that you do this. Otherwise, you will find yourself severely handicapped on tests.
Here is a unit circle with popular angle values on it:
enter image source here
You can find many versions of the unit circle online to print for your use.
The values you saw in my solution came from the unit circle.