Trigonometric Identities, I don't know where to start (number 4)

[James Ibarra]

Hi, I'm an 11th grade student having problems on pre calculus
Our instructorr gave us this exercise to atleast have an idea of what trigonometric identities look like because we were about to discuss this topic but due covid19 we were not able to. Can someone guide me into understanding this? Thanks!

 

[James Ibarra]

can anybody help me with some of these, I got up to number 3 then got stuck on number 4.

 

[MarkFL]

can anybody help me with some of these, I got up to number 3 then got stuck on number 4.

Hello again!

I've created a new thread for your question about number 4. Let's look at it:

[MATH](\csc(\theta)-\cot(\theta))^2[/MATH]
Let's combine within the brackets and write as:

[MATH]\frac{(1-\cos(\theta))^2}{\sin^2(\theta)}[/MATH]
Using a Pythagorean identity we may write:

[MATH]\frac{(1-\cos(\theta))^2}{1-\cos^2(\theta)}[/MATH]
Can you proceed?

 

[Deleted member 4993]

View attachment 18130

Hi, I'm an 11th grade student having problems on pre calculus
Our instructorr gave us this exercise to atleast have an idea of what trigonometric identities look like because we were about to discuss this topic but due covid19 we were not able to. Can someone guide me into understanding this? Thanks!

Express \(\displaystyle csc(\theta) \ \ and\ \ cot (\theta)\) in terms of \(\displaystyle cos(\theta) \ \ and\ \ sin (\theta)\) and simplify.

If you are still stuck, please show your work and tell us exactly where you are stuck.

 

[James Ibarra]

you substituted csc(θ) with 1/sin(θ) right? how did the sin in the denominator squared? sorry if I'm asking dumb questions.
I understand the concept now, but I'm still having trouble with operations like how did it become this and where did this come from.
by continuing this equation (1-cos(θ))^2/1-cos^2(θ) i factor out the denominator to positive and negative to cancel out like terms and end up with the answer letter K.
Do you sometimes feel when working with equations you get the correct answer but you have no idea what you're doing? i feel like that right now.

 

[James Ibarra]

Express \(\displaystyle csc(\theta) \ \ and\ \ cot (\theta)\) in terms of \(\displaystyle cos(\theta) \ \ and\ \ sin (\theta)\) and simplify.

If you are still stuck, please show your work and tell us exactly where you are stuck.

I'm particularly having trouble with number 8 now. literally don't know how to tackle this equation.

 

[James Ibarra]

I talked with my instructor, he told me that some of these equation required the used of complex identities.

 

[James Ibarra]

you substituted csc(θ) with 1/sin(θ) right? how did the sin in the denominator squared? sorry if I'm asking dumb questions.
I understand the concept now, but I'm still having trouble with operations like how did it become this and where did this come from.
by continuing this equation (1-cos(θ))^2/1-cos^2(θ) i factor out the denominator to positive and negative to cancel out like terms and end up with the answer letter K.
Do you sometimes feel when working with equations you get the correct answer but you have no idea what you're doing? i feel like that right now.

welp, this reply was completely dumb, i subconciously switched cot the with cos and now I'm back to square one. haha

 

[MarkFL]

you substituted csc(θ) with 1/sin(θ) right?

Yes, inside the brackets, I used:

[MATH]\csc(\theta)-\cot(\theta)=\frac{1}{\sin(\theta)}-\frac{\cos(\theta)}{\sin(\theta)}=\frac{1-\cos(\theta)}{\sin(\theta)}[/MATH]
And this is of course squared:

[MATH]\left(\frac{1-\cos(\theta)}{\sin(\theta)}\right)^2[/MATH]
Now, in algebra, we know:

[MATH]\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}[/MATH]
So, we can write:

[MATH] \frac{(1-\cos(\theta))^2}{\sin^2(\theta)}[/MATH]

sorry if I'm asking dumb questions.

The only dumb questions are the ones left unasked, when you're trying to understand something.

understand the concept now, but I'm still having trouble with operations like how did it become this and where did this come from.
by continuing this equation (1-cos(θ))^2/1-cos^2(θ) i factor out the denominator to positive and negative to cancel out like terms and end up with the answer letter K.

Yes, that's correct.

Do you sometimes feel when working with equations you get the correct answer but you have no idea what you're doing? i feel like that right now.

When I was a student, and presented with verifying trig. identities, it was the most difficult thing I had experience at that point. Even now they can be tricky.

I'm particularly having trouble with number 8 now. literally don't know how to tackle this equation.

[MATH]\sin\left(\theta-\frac{\pi}{6}\right)+\cos\left(\theta-\frac{\pi}{3}\right)[/MATH]
Are you familiar with the angle sum/difference identities for sine and cosine?

 

[James Ibarra]

Yes, inside the brackets, I used:

[MATH]\csc(\theta)-\cot(\theta)=\frac{1}{\sin(\theta)}-\frac{\cos(\theta)}{\sin(\theta)}=\frac{1-\cos(\theta)}{\sin(\theta)}[/MATH]
And this is of course squared:

[MATH]\left(\frac{1-\cos(\theta)}{\sin(\theta)}\right)^2[/MATH]
Now, in algebra, we know:

[MATH]\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}[/MATH]
So, we can write:

[MATH] \frac{(1-\cos(\theta))^2}{\sin^2(\theta)}[/MATH]


The only dumb questions are the ones left unasked, when you're trying to understand something.



Yes, that's correct.



When I was a student, and presented with verifying trig. identities, it was the most difficult thing I had experience at that point. Even now they can be tricky.



[MATH]\sin\left(\theta-\frac{\pi}{6}\right)+\cos\left(\theta-\frac{\pi}{3}\right)[/MATH]
Are you familiar with the angle sum/difference identities for sine and cosine?

Thank you for your help mark, sorry for replying so late in the day, but with your help I was able to answer all the questions. Even though I shed a tear and spaced out a couple of times thinking of what to do next I was still able to pull through. Thanks to this forum I learned something in 2 days that would take a week if discussed in school.


I also would like to ask if there's a limit on how many threads i can make? Because after I finished the identities, I did the next part of the homework which involves triangles, and I got through them but I was confused on a particular question and would like to share it with members of these forums, so I may know if I got the correct answer or not.


But anyways, THANK YOU SO MUCH FOR YOUR HELP MARK! I'm really grateful! Godbless and Staysafe during the pandemic!

 

[Deleted member 4993]

Thank you for your help mark, sorry for replying so late in the day, but with your help I was able to answer all the questions. Even though I shed a tear and spaced out a couple of times thinking of what to do next I was still able to pull through. Thanks to this forum I learned something in 2 days that would take a week if discussed in school.


I also would like to ask if there's a limit on how many threads i can make? Because after I finished the identities, I did the next part of the homework which involves triangles, and I got through them but I was confused on a particular question and would like to share it with members of these forums, so I may know if I got the correct answer or not.


But anyways, THANK YOU SO MUCH FOR YOUR HELP MARK! I'm really grateful! Godbless and Staysafe during the pandemic!

There is no limit on number of threads. Just make sure that the thread subject-lines are unique. You may start as many threads as you can - with one problem/thread.