Solving a multiple of cosine of an angle in exact terms

[lowellie]

Hello,

I'm stuck on how to solve/simplify a number multiplied by cosine of an angle.

Original problem: I'm given an angle (330 degrees) and an absolute value (4). And I need to find a and b, then express the answer in exact terms.

How far I got: Following SOH CAH TOA, I set up an equation for a, and one for b.

cos(330)=a/4 (multiplied 4 on both sides)
4cos(330)=a

sin(330)=b/4 (multiplied 4 on both sides)
4sin(330)=b

From here, I know the reference angle is 360-330=30 degrees and that cosine represents the x value. If there was no "4" in front of the problem, I know to use the reference angle of 30 degrees and find the x value, which would be the square root of 3 over 2. But how do I solve the problem when there is a number in front of cosine (or sine)? Please help and thanks so much for your time.

 

[Deleted member 4993]

Hello,

I'm stuck on how to solve/simplify a number multiplied by cosine of an angle.

Original problem: I'm given an angle (330 degrees) and an absolute value (4). And I need to find a and b, then express the answer in exact terms.

How far I got: Following SOH CAH TOA, I set up an equation for a, and one for b.

cos(330)=a/4 (multiplied 4 on both sides)
4cos(330)=a

sin(330)=b/4 (multiplied 4 on both sides)
4sin(330)=b

From here, I know the reference angle is 360-330=30 degrees and that cosine represents the x value. If there was no "4" in front of the problem, I know to use the reference angle of 30 degrees and find the x value, which would be the square root of 3 over 2. But how do I solve the problem when there is a number in front of cosine (or sine)? Please help and thanks so much for your time.

What do you get when you "divide" one equation by the other?

 

[lowellie]

Hello,

Thanks for your response. But I'm not sure about what you're asking. Let's see, if I were to divide one equation by the other...

4cos(330)/4sin(330) ... wouldn't that just leave me with cos/sin?

I thought I had to solve each separately and write it in the form a+bi, but in exact terms. Or is the dividing by each other a way to get there?

 

[Dr.Peterson]

Hello,

I'm stuck on how to solve/simplify a number multiplied by cosine of an angle.

Original problem: I'm given an angle (330 degrees) and an absolute value (4). And I need to find a and b, then express the answer in exact terms.

How far I got: Following SOH CAH TOA, I set up an equation for a, and one for b.

cos(330)=a/4 (multiplied 4 on both sides)
4cos(330)=a

sin(330)=b/4 (multiplied 4 on both sides)
4sin(330)=b

From here, I know the reference angle is 360-330=30 degrees and that cosine represents the x value. If there was no "4" in front of the problem, I know to use the reference angle of 30 degrees and find the x value, which would be the square root of 3 over 2. But how do I solve the problem when there is a number in front of cosine (or sine)? Please help and thanks so much for your time.

I think it would be easier to help you if we saw the actual exact problem as given to you, and also knew a little more about what you have learned so far. I think you are right at the beginning of learning about trigonometry ... no, after looking at your more recent response, it looks like the context is learning about the polar form of complex numbers! We really do need to see your context!

But my impression is that you just need to find the exact value of the sine and cosine, as if that were the entire problem, and then multiply those results by 4.

 

[lowellie]

Hello Dr. Peterson,

Sorry for any confusion, but what I initially wrote was the exact problem given to me: "Original problem: I'm given an angle (330 degrees) and an absolute value (4). And I need to find a and b, then express the answer in exact terms." And "in the form a+bi."

So, if I find the exact value of sine and cosine, then multiply those results by 4, I would get:

4cos(330)=a --> 4 times the square root of 3 over 2

4sin(330)=b --> 4 times negative 1/2

Then that would give me ... 4 times the square root of 3 over 2 minus 2i

Is that correct?

 

[canvas]

[math]\text{Just use this formulas!}\\\cos(360\degree-\alpha)=\cos\alpha\\\\\sin(360\degree-\alpha)=-\sin\alpha[/math]

 

[Dr.Peterson]

Sorry for any confusion, but what I initially wrote was the exact problem given to me: "Original problem: I'm given an angle (330 degrees) and an absolute value (4). And I need to find a and b, then express the answer in exact terms." And "in the form a+bi."

No problem in a textbook would ever read that way; you didn't quote it exactly, as we ask. But at the least, you omitted some very important information; in what you originally wrote, a and b weren't defined at all, and there was no indication that these were the angle and absolute value of a complex number.

But now we know.

4cos(330)=a --> 4 times the square root of 3 over 2

4sin(330)=b --> 4 times negative 1/2

Then that would give me ... 4 times the square root of 3 over 2 minus 2i

Is that correct?

Yes, though you can simplify it to [imath]2\sqrt{3} - 2i[/imath].

To type what you wrote, you could say "4 sqrt(3)/2 - 2i" rather than using words.

 

[lowellie]

Ah, okay, I was tripped up at seeing the 4 in 4cos(330). I thought I had to do something with 4 AND cosine. I didn't realize that I calculate cos(330) first, then simplify 4 sqrt(3)/2. I'm just learning, so thanks for bearing with me and for all your help.

 

[Harry_the_cat]

Ah, okay, I was tripped up at seeing the 4 in 4cos(330). I thought I had to do something with 4 AND cosine. I didn't realize that I calculate cos(330) first, then simplify 4 sqrt(3)/2. I'm just learning, so thanks for bearing with me and for all your help.

Yes \(\displaystyle 4cos(330^o) = 4 * cos(330^o)\).