# Unit Circle Help

• May 15th 2012, 04:43 AM
ineedhelplz
Unit Circle Help
Hi, I am currently tutoring yr 11 and 12.

I had difficulty explaining how to determine unit circle values without the use of known formulas

For example:

tan(3*pi/2 - a) = cot(a) OR sin(pi/2 - b) = cos(b)

How can I explain it with just the unit circle - not the given equivalencies?

Thanks
• May 15th 2012, 05:44 AM
Prove It
Re: Unit Circle Help
Quote:

Originally Posted by ineedhelplz
Hi, I am currently tutoring yr 11 and 12.

I had difficulty explaining how to determine unit circle values without the use of known formulas

For example:

tan(3*pi/2 - a) = cot(a) OR sin(pi/2 - b) = cos(b)

How can I explain it with just the unit circle - not the given equivalencies?

Thanks

The second is easier to show using any right-angle triangle.
• May 15th 2012, 07:47 AM
Deveno
Re: Unit Circle Help
using "radian" notation, we have that the sum of the 3 angles of any triangle is equal to π. if our triangle is a right triangle, then our "main angle" (the adjacent angle, formed by the hypotenuse and the x-axis) and the "other angle" (the opposite angle, formed by the hypotenuse and the vertical leg) sum to π/2.

if we "swap" the angles, we turn the vertical leg into a horizontal one, and the x-axis becomes the y-axis (imagine taking that same triangle, and moving the point that was at the center of the unit circle to the outer radius, while moving the point that was on the circle radius to the origin, and then "rotating and flipping" to switch the axes).

this means that the side that was the cosine, is now the sine of the opposite angle, and that the side that was the sine, is now the cosine of the opposite angle. so

sin(π/2 - θ) = cos(θ)

cos(π/2 - θ) = sin(θ)
• May 15th 2012, 02:50 PM
ineedhelplz
Re: Unit Circle Help
Thank you very much for that, is there a visual aid you could please show me?

Thanks!
• May 15th 2012, 06:38 PM
Prove It
Re: Unit Circle Help
http://i22.photobucket.com/albums/b3...agon/RAT-1.jpg

I'm sure you can see

\displaystyle \begin{align*} \sin{\left(\frac{\pi}{2} - \theta\right)} &= \frac{b}{h} \\ \\ \cos{\theta} &= \frac{b}{h} \\ \\ \sin{\left(\frac{\pi}{2} - \theta\right)} &= \cos{\theta} \end{align*}
• May 16th 2012, 03:12 AM
ineedhelplz
Re: Unit Circle Help
I understand that thank you, however what about the first example?

tan(3pi/2 - a) or
tan(3pi/2 + a)

I don't see a way to do that with a right angled triangle. Wouldn't you need the unit circle?
• May 16th 2012, 03:19 AM
ineedhelplz
Re: Unit Circle Help
The only way I see is to make equivalencies with the unit circle.

For example tan(3pi/2 + a) = -tan(pi/2 - a)

Is there any way to demonstrate this with only the unit circle?

Thanks again