I/D #1 - Unit N Concept 7: How do SRTs & the UC relate?

Inquiry Activity Summary:
The unit circle activity is to help us develop our understanding of the special right triangles and how it correlates with the unit circle. There are three special right triangles: 30º, 45º, and 90º. The three sides of the triangle has to be simplified in such a way that the hypotenuse is equal to one because when inserting the triangle into the unit circle, the hypotenuse would be the radius, which always have to be one.

1. Rules for 30º triangle:

• Hypotenuse (r) is 2x *Remember to change it to 1 by dividing 2x*  
• Horizontal value (x) is x *When divided by 2x, it will make 1/2* 
• Vertical value (y) is x radical 3 *When divided by 2x, it will be radical 3/2*
• The 30º angle must be at the origin (0,0)

2. Rules for 45º triangle:

• Hypotenuse is radical 2 but remember to divide by radical 2 to make it 1
• Horizontal value is 1 over radical 2, when divided it'll be radical 2 /2
• Vertical value will be the same as horizontal value
• 45º angle must be at origin

3. Rules for 60º triangle:

• Hypotenuse is 2x, divide by 2x and it'll become 1
• Horizontal value is x radical 3, after dividing it'll be radical 3 / 2
• Vertical value is x, after dividing it'll be 1/2
• 60º must be at origin

4. The special right triangles activity helps us derive the unit circle because the points from the triangles will form a circle if all the points are connected. But of course, the 30º, 45º, & 60º are only in the first quadrant. So, you would have to find the other coordinate pairs, however, once you know quadrant 1, you basically know all the other quadrants.

5. The triangles drawn in this activity lies in quadrant 1. The values will change with positive or negative depending on which quadrant you're in. If you're in the 2nd quadrant, then your x value will be negative and your y value is positive. If it's in the 3rd, both the x and y value will be negative. If it's in the 4th, x value will be positive and y value negative.

In this 2nd quadrant, I showed the 30º reference angle,
the actual angle is 150º. As you can see, the coordinate pair
farthest to the right, is similar to the 30º angle in quadrant I.
The only difference is the x value is negative.

In this 3rd quadrant, I showed the 45º reference angle, the angle
is 225º. The coordinate pair closest to the bottom has both a negative
y and x value. It's also similar to 45º in quadrant I.

In quadrant IV, I showed the 60º reference angle and the angle
would be 300º. The coordinate pair closest to the bottom has a negative
y value and a positive x value. This angle is similar to the 60º angle
in quadrant I.

Inquiry Activity Reflection:

"The coolest thing I learned from this activity was..." The unit circle consists of special right triangles and it's not just a bunch of numbers you have to memorize; there's a pattern.

"This activity will help me in this unit because..." I'll be able to comprehend the unit circle without having to memorize random points.

"Something I never realized before about special right triangles and the unit circle is..." That just by knowing the first quadrant, you pretty much memorized the entire unit circle.