Complimentary Angles: Understand Why sin(90-theta) = cos theta

[Miike012]

Homework Statement

It bascially says when the sum of two angles are 90 deg they are complimentary...
Then it goes on saying
sin(90-theta) = cos theta
and so on...

I understand the concept such as if sin(30) = cos(60)
But the book doesn't explain why it works.. it just says if two compl angles are 90 deg then sin(90-theta) = cos theta, which is not an explanation at all...
can some one help me understand why this is?

Thank you.

Homework Equations

The Attempt at a Solution

 

[eumyang]

I understand the concept such as if sin(30) = cos(60)

First of all, you forgot the degree marks. If you write "sin(30) = cos(60)", I'm assuming you meant radian measure. You must write sin (30°) = cos (60°). Secondly, If you "understood" the concept above, then why are you asking it in the first place? Sounds like you really don't understand.

Look at this diagram:
[PLAIN]http://home.comcast.net/~yeongil/images/RtTri.jpg
In a right triangle, the two non right-angles have to be complements of each other.
$$\sin B = \frac{opp}{hyp} = \frac{b}{c}$$
and
$$\cos A = \frac{adj}{hyp} = \frac{b}{c}$$
so
$$\sin B = \cos A$$.
Since B = 90° - A,
$$\sin (90^{\circ} - A) = \cos A$$.