A few Trig questions about radians/input/theory/etc. A beginner.

Jan 2011
10
0
Okay I've been trying my hardest to get the grasp of Trigonometry down, but I think I have overwhelmed myself. I have a few questions and I am going to try to word them so that they are not complicated and won't clutter up the thread.

1. What exactly is Sine (or Cosine)? The reason I ask, is because I was taught two things. One, that Sine is the Y-value on the Unit Circle or Graph. Secondly, that Sine is the ratio of Opposite/Hypotenuse in a Right Triangle. This confuses me because one seems to be a ratio and another a value (in Radians). I'm sure I've confused myself over nothing.

2. When I enter a value, such as X = -500 for Sine (y = Sine x), what does this value actually mean? Does it mean -500 degrees? Or -500 Radians? On a calculator, I plugged in -500 sine (in degrees) and it gives me -.64. -500 sine (in Radians) gives me .46. What do those values correspond to and why are the signs different? I assume they are the point on the Y-axis, but why are they different values and signs?

I guess I will stop there, since the answers to these may answer my other questions. Thanks to anyone that helps me!
 

dwsmith

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Mar 2010
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\(\displaystyle \displaystyle\sin{\theta}=\frac{y}{r}\)

\(\displaystyle y=r\sin{\theta}\)

Sine is the y value if r = 1
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
2. When I enter a value, such as X = -500 for Sine (y = Sine x), what does this value actually mean? Does it mean -500 degrees? Or -500 Radians? On a calculator, I plugged in -500 sine (in degrees) and it gives me -.64. -500 sine (in Radians) gives me .46. What do those values correspond to and why are the signs different? I assume they are the point on the Y-axis, but why are they different values and signs?

I guess I will stop there, since the answers to these may answer my other questions. Thanks to anyone that helps me!
\(\displaystyle \displaystyle\mbox{Radian}\frac{\pi}{2}=\mbox{Degree} \ 90^{\circ}\)

360 degrees is 2pi radians.

It gives you the location in either radians or degrees. For instance, pi radians is 180 degrees.
 
Jan 2011
10
0
Ok that helped me a little. So I am trying to work all this out by hand, so how does this work. If I have an angle that is -140 degrees, how do I manually figure out the Sine and Cosine on the unit circle? I drew a picture, but since I don't know the length of X or Y, I don't know how to proceed, all I have is the Angles and a radius of 1.

I drew a right triangle with a radius of 1 and angles 40 and 50 degrees inside. I'm not sure how to find Y or X for the triangle. I assume I HAVE to use a calculator or the calculation would take too long?
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
-140 can be re-written as 220 degrees

\(\displaystyle \displaystyle\sin{220^{\circ}}=\frac{y}{r}\Rightarrow \sin{220^{\circ}}=\frac{y}{1}\Rightarrow y=\sin{220^{\circ}}\)

\(\displaystyle \displaystyle\cos{220^{\circ}}=\frac{x}{r}\Rightarrow \cos{220^{\circ}}=\frac{x}{1}\Rightarrow x=\cos{220^{\circ}}\)
 
Jan 2011
10
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Ok I worked out two problems, one with Sine (-500) in Degrees and one with Sine (-500) in Radians. I am just doing this to help me understand all the concepts, I will end up using a calculator for everything once I get it.

1. Sine (-500) in Degrees gave me a Sine of -.64 and Cosine of -.766. So this is in Quadrant III. Also, the angle of measurement would be -500 degrees, -140 Degrees, or 220 degrees.

2. Sine (-500) in Radians gave me a Sine of .46 and Cosine of .883. So this is in Quadrant I. The angle of measurement would be -28, 647.27 degrees or 27.38 degrees.
 

dwsmith

MHF Hall of Honor
Mar 2010
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Florida
-500+360+360=220

Angle -500 = -140 = 220 = 580 = .....

\(\displaystyle \theta+360k \ \ k\in\mathbb{Z}\)

\(\displaystyle \theta+\pi k\ \ k\in\mathbb{Z}\)

You can't take the since of -500 degrees in radians. You need to take it in degrees.

-500 radians isn't the same as -500 degrees.
 
Jan 2011
10
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I thought calculating -500 on a calculator (using the Radian function) calculated the Sine of that value at -500 Radians.
 

dwsmith

MHF Hall of Honor
Mar 2010
3,093
582
Florida
I thought calculating -500 on a calculator (using the Radian function) calculated the Sine of that value at -500 Radians.
It does but isn't the -500 degrees in your original post?
 
Jan 2011
10
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Oh I was just using the same number (for Degrees and Radians) to see the difference in the answers and ways of working them.