How can someone prove the following trigonometric identity?

In summary, the conversation revolves around finding a way to prove the equation C = Sqrt[A^2 + B^2] and theta = arctan(B/A). The suggested method involves using geometry and trigonometric identities to create triangles that demonstrate the validity of the equation.
  • #1
cocopops12
30
0
it's bothering my brain..i thought about it many times...i can't make intuition of it
can anyone prove it?
9qz7u8.png


oh by the way... C = Sqrt[A^2 + B^2] and theta is equal to arctan(B/A)
 
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  • #2
Hey cocopops12 and welcome to the forums.

The easiest way IMO is to use geometry.

Think about for example right angled triangle with the hypotenuse coordinates (Bsin(wt),Acos(wt)).

Now think about the angles and sides involved where in your RHT, you have (Acos(wt))^2 + (Bsin(wt))^2 = hypotenuse.
 
  • #3
Perhaps you could use the angle-difference-identity of the cosine?
See: http://en.wikipedia.org/wiki/Trig_identities#Angle_sum_and_difference_identities

That is: ##\cos(ωt-θ)=\cos(ωt)\cos(θ) + \sin(ωt)\sin(θ)##.

@chiro: I don't understand how your triangle would work out...
However, I can think of different triangles that would fit.
You can draw 3 rectangular triangles with sides (A,B,√(A2+B2)), (Acos(ωt),Asin(ωt),A), and (Bcos(ωt),Bsin(ωt),B).
In the proper configuration these 3 triangles show the result.
 
Last edited:
  • #4
I like Serena said:
Perhaps you could use the angle-difference-identity of the cosine?
See: http://en.wikipedia.org/wiki/Trig_identities#Angle_sum_and_difference_identities

That is: ##\cos(ωt-θ)=\cos(ωt)\cos(θ) + \sin(ωt)\sin(θ)##.

@chiro: I don't understand how your triangle would work out...
However, I can think of different triangles that would fit.
You can draw 3 rectangular triangles with sides (A,B,√(A2+B2)), (Acos(ωt),Asin(ωt),A), and (Bcos(ωt),Bsin(ωt),B).
In the proper configuration these 3 triangles show the result.

Yeah that would be optimal.
 
  • #5


Proving trigonometric identities can be a challenging task, especially when they seem counterintuitive. However, there are several methods that can be used to prove a trigonometric identity.

One method is to use algebraic manipulation and substitution to simplify the expression on both sides of the identity until they are equivalent. This can involve using trigonometric identities, basic algebraic rules, and properties of special angles.

Another method is to use geometric proofs, which involve drawing a diagram and using geometric properties and theorems to show that the two sides of the identity represent the same geometric concept.

It is also important to note that some identities may require a combination of both algebraic and geometric proofs to fully prove.

In the case of the given identity, it may be helpful to start by substituting the given values for C and theta into the expression and then simplifying from there. Additionally, drawing a right triangle and labeling the sides with the given values may provide insight into how the identity can be proven geometrically.

Overall, proving trigonometric identities requires patience, careful reasoning, and a thorough understanding of trigonometric concepts and properties. It may take some time and effort, but with persistence, the identity can be proven.
 

1. How do I approach proving a trigonometric identity?

Proving a trigonometric identity involves using mathematical manipulations and algebraic techniques to show that both sides of the equation are equivalent. It is important to start with a clear understanding of the identities and properties of trigonometric functions.

2. What are some common strategies for proving trigonometric identities?

Some common strategies for proving trigonometric identities include using basic trigonometric identities, simplifying expressions using algebraic techniques, and transforming one side of the equation to match the other.

3. How do I know if I have proven a trigonometric identity correctly?

To ensure that your proof is correct, you can check that both sides of the equation are equivalent by substituting different values for the variables and checking if the equation holds true for all values. You can also verify your proof with a calculator or using online tools.

4. Can I use identities to prove other identities?

Yes, you can use identities that have already been proven to help prove other identities. This is known as the transitive property of equality.

5. How can I practice and improve my skills in proving trigonometric identities?

You can practice by solving different types of trigonometric identities and equations, starting with simple ones and gradually moving on to more complex ones. You can also seek help from online resources, textbooks, or a math tutor to improve your skills.

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