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

In summary, the conversation discusses a trigonometric identity involving angles and sides of two equal triangles, and the importance of setting the sides equal in order to prove the identity. It also explains why the identity would still hold even if the triangles were not equal in all respects. Finally, it addresses the concept of sine and how it relates to the sides of a triangle.
  • #1
Miike012
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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

 

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  • #2
Miike012 said:
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.
Miike012 said:
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

[tex]\Delta POM \equiv \Delta OP'M'[/tex]

Miike012 said:
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.

Miike012 said:
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 [tex]\sin \theta[/tex] is equal to in terms of the sides in the triangle OPM.
 
  • #3
Sin Theta in terms of Triangle OPM would be... MP/OP
 

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

1. What is a trig identity?

A trig identity is a mathematical equation that is true for all possible values of the variables involved. It is a way of expressing a relationship between trigonometric functions that is always true, regardless of the specific values of the angles involved.

2. Why are trig identities important?

Trig identities are important because they allow us to simplify complex trigonometric expressions, making them easier to work with and manipulate. They also help us solve trigonometric equations and prove other mathematical theorems.

3. How do I prove a trig identity?

There are several ways to prove a trig identity, including using algebraic manipulations, using geometric interpretations, and using the properties of trigonometric functions. It is important to have a solid understanding of the fundamental trigonometric identities and their properties in order to successfully prove a trig identity.

4. What are some common trig identities?

Some common trig identities include the Pythagorean identities (sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ), the double angle identities (sin(2θ) = 2sinθcosθ, cos(2θ) = cos²θ - sin²θ, and tan(2θ) = 2tanθ / (1 - tan²θ)), and the half angle identities (sin(θ/2) = ±√[(1 - cosθ) / 2], cos(θ/2) = ±√[(1 + cosθ) / 2], and tan(θ/2) = sinθ / (1 + cosθ)).

5. How can I use trig identities in real-world applications?

Trig identities are used in a variety of real-world applications, such as engineering, physics, and navigation. For example, the Pythagorean identities are used in calculating forces and distances in physics, and the double angle identities are used in calculating the period of a pendulum. In navigation, trig identities are used to calculate distances and angles between points on a map or globe.

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