Solving Identifying $\theta$ in Trigonometric Equation

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In summary, the conversation discusses how to prove the identity $\frac{\csc\theta+1}{\cot\theta}=\frac{\cot\theta}{\csc\theta-1}$ using various trigonometric identities. The approach involves using the fundamental identity $1+\cot^{2}\theta=\csc^{2}\theta$ and factoring using the Difference of Two Squares pattern. There are some typos in the equations due to typing on a phone, but the basic idea is correct.
  • #1
paulmdrdo1
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Kindly help me with this problem. I'm stuck!

$\frac{\csc\theta+1}{\cot\theta}=\frac{\cot\theta}{\csc\theta-1}$

this is how far I can get to

$\sec\theta+\tan\theta=\frac{1}{\sec\theta-\tan\theta}$
 
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  • #2
paulmdrdo said:


Kindly help me with this problem. I'm stuck!

$\frac{\csc\theta+1}{\cot\theta}=\frac{\cot\theta}{\csc\theta-1}$

this is how far I can get to

$\sec\theta+\tan\theta=\frac{1}{\sec\theta-\tan\theta}$


I would start with
$$\sin^2(\theta)+\cos^2(\theta)=1,$$
divide through by $\sin^2(\theta)$, and see what happens.
 
  • #3
Ackbach said:
I would start with
$$\sin^2(\theta)+\cos^2(\theta)=1,$$
divide through by $\sin^2(\theta)$, and see what happens.

I would have this fundamental identity

$1+\cot^{2}\theta=\csc^{2}\theta$ where should I put this?
 
  • #4
paulmdrdo said:
I would have this fundamental identity

$1+\cot^{2}\theta=\csc^{2}\theta$ where should I put this?

Right. So when I see the $\csc(\theta)-1$ on one side of the equation (I mean the identity you're trying to prove), and a $\csc(\theta)+1$ on the other side of the equation, my thoughts immediately go to the Difference of Two Squares factoring pattern. Can you see what might need to happen here?
 
  • #5
Ackbach said:
Right. So when I see the $\csc(\theta)-1$ on one side of the equation (I mean the identity you're trying to prove), and a $\csc(\theta)+1$ on the other side of the equation, my thoughts immediately go to the Difference of Two Squares factoring pattern. Can you see what might need to happen here?

Here it is

Using this identity $\cot^{2}\theta=(\csc\theta+1)(\csc\theta-1)$

$(\csc\theta+1)=\frac{\cot^{2}\theta}{\csc\theta-1}$

$\frac{\cot^{2}\theta}{\cot(\csc\theta-1)}=\frac{\cot\theta}{\csc\theta-1}$

$\frac{\cot\theta}{\csc\theta-1}=\frac{\cot\theta}{\csc\theta-1}$
 
  • #6
paulmdrdo said:
Here it is

Using this identity $\cot^{2}\theta=(\csc\theta+1)(\csc\theta-1)$

$(\csc\theta+1)=\frac{\cot^{2}\theta}{\csc\theta-1}$

$\frac{\cot^{2}\theta}{\cot(\csc\theta-1)}=\frac{\cot\theta}{\csc\theta-1}$

You've the basic idea right, but you don't want to write $\cot(\csc(\theta)-1)$. Instead, write $\cot(\theta)(\csc(\theta)-1)$.

$\frac{\cot\theta}{\csc\theta-1}=\frac{\cot\theta}{\csc\theta-1}$

You need to clean up your equations - there are some typos in there - but I think you've got the basic idea.
 
  • #7
Ackbach said:
You've the basic idea right, but you don't want to write $\cot(\csc(\theta)-1)$. Instead, write $\cot(\theta)(\csc(\theta)-1)$.
You need to clean up your equations - there are some typos in there - but I think you've got the basic idea.

I'm using my phone right now. That is why my typing is messed up. Thanks for your help though.
 
  • #8
paulmdrdo said:
Thanks for your help though.

You're very welcome!
 

FAQ: Solving Identifying $\theta$ in Trigonometric Equation

What is the purpose of solving for $\theta$ in a trigonometric equation?

The purpose of solving for $\theta$ in a trigonometric equation is to find the unknown angle in a given triangle or other geometric shape. This allows us to accurately measure and describe the shape and its properties.

What are the common methods used to solve for $\theta$ in trigonometric equations?

The common methods used to solve for $\theta$ in trigonometric equations include using trigonometric identities, the unit circle, and the inverse trigonometric functions sine, cosine, and tangent.

How do I know which method to use when solving for $\theta$?

The method used to solve for $\theta$ depends on the given equation and the available information. For example, if the equation involves only trigonometric functions, identities may be used. If the equation involves a right triangle, the unit circle or inverse trigonometric functions may be used.

What are some tips for solving for $\theta$ in trigonometric equations?

Some tips for solving for $\theta$ in trigonometric equations include carefully examining the given information, using appropriate trigonometric identities, and being familiar with the unit circle and inverse trigonometric functions.

Can I use a calculator to solve for $\theta$ in a trigonometric equation?

Yes, a calculator can be used to solve for $\theta$ in a trigonometric equation. Many calculators have a built-in function for inverse trigonometric functions, making it easier to solve for $\theta$. However, it is important to understand the steps involved in solving for $\theta$ manually in order to use the calculator effectively.

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