1. ## 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

2. ## Re: Unit Circle Help

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.

3. ## 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)

4. ## Re: Unit Circle Help

Thank you very much for that, is there a visual aid you could please show me?

Thanks!

5. ## Re: Unit Circle Help

I'm sure you can see

\displaystyle \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*}

6. ## 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?

7. ## 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