prove sinQ=cos(90-Q)
Answers
Answer:
Explanation:
Answer:
I prefer a geometric proof. See below.
Explanation:
If you're looking for a rigorous proof, I'm sorry - I'm not good at those. I'm sure another Socratic contributor like George C. could do something a little more solid than I can; I'm just going to give the lowdown on why this identity works.
Take a look at the diagram below:
enter image source here
It's a generic right triangle, with a
90
o
angle as indicated by the little box and an acute angle
a
. We know the angles in a right triangle, and a triangle in general, must add to
180
o
, so if we have an angle of
90
and an angle of
a
, our other angle must be
90
−
a
:
(
a
)
+
(
90
−
a
)
+
(
90
)
=
180
180
=
180
We can see that the angles in our triangle do indeed add to
180
, so we're on the right track.
Now, let's add some variables for side length onto our triangle.
enter image source here
The variable
s
stands for the hypotenuse,
l
stands for length, and
h
stands for height.
We can start on the juicy part now: the proof.
Note that
sin
a
, which is defined as opposite (
h
) divided by hypotenuse (
s
) , equals
h
s
in the diagram:
sin
a
=
h
s
Note also that the cosine of the top angle,
90
−
a
, equals the adjacent side (
h
) divided by the hypotenuse (
s
):
cos
(
90
−
a
)
=
h
s
So if
sin
a
=
h
s
, and
cos
(
90
−
a
)
=
h
s
...
Then
sin
a
must equal
cos
(
90
−
a
)
!
sin
a
=
cos
(
90
−
a
)
And boom, proof complete.