Why does Sin (right angle plus theta) equal cos(theta)?

[Miike012]

Homework Statement

I am reading an explanation on a trig identity but I am not fully understanding it...

Angle MOP = Theta
Angle POP' = Right angle
Angle AOP' = (Theta + Right angle)
Take OP' = OP ( WHY must it be equal?)
Angle MOP + P'OM' = 90 ( I understand this)
Angle MOP = Angle OP'M' (I understand this)
Both Triangles MOP And M'P'O are equal in all respects (What makes them equal ? Is it because they both have one side equal and an angle that is also equal? And how is this Important to the theorem?)

If these triangle were not "equal in all respects," would this theorem not work?

Then it says... Sin ( right angle plus theta) = cos(theta)
and so on...

I don't understand why Sin ( right angle plus theta) = cos(theta)

Can some one please explain...

Homework Equations

The Attempt at a Solution

 

[Mentallic]

Take OP' = OP ( WHY must it be equal?)

We set them equal. We did it on purpose. It doesn't have to be equal but it makes the triangles equal further down, rather than just similar.

Both Triangles MOP And M'P'O are equal in all respects (What makes them equal ? Is it because they both have one side equal and an angle that is also equal? And how is this Important to the theorem?)

Because of the Angle-Angle-Side rule (AAS). We have OP=OP', angle POM = angle OP'M', angle OM'P' = angle OMP. Thus the triangles are equivalent. You would need to write the triangle sides in such a way that the corners match up equally. We would write

$$\Delta POM \equiv \Delta OP'M'$$

If these triangle were not "equal in all respects," would this theorem not work?

The simple answer is no it would still work. The sine of an angle doesn't depend on the size of the triangle. They are only ratios, not exact lengths. So if the sides of the triangles weren't equal, they would still be similar (their angles equal) and the theorem would still state the same thing. Making OP=OP' just makes it easier to understand and follow.

Then it says... Sin ( right angle plus theta) = cos(theta)
and so on...

I don't understand why Sin ( right angle plus theta) = cos(theta)

Can some one please explain...

You agree the triangles are equal based on our constructions right? Such as OP=OP'. We have proved they are equal. Ok, so tell me what $$\sin \theta$$ is equal to in terms of the sides in the triangle OPM.

 

[Miike012]

Sin Theta in terms of Triangle OPM would be... MP/OP