If This Triangle Is 45-45-90, What Are the Other Two Sides in Terms of X

Given the following arc lengths of a unit circle determine a way to find the vertical and horizontal displacements from the center of the circle for the endpoint of the arc. Round your answers for these distances to the nearest hundredth. If the arc of length of a circle is s, the angle  in radians, and r the radius. The equation is: s= r q

ALSO: In a circle there are 2p radians in a full circle. Thus 360 degrees is about 6.28… radians. Use the exact to convert.

1 (cos(), sin())

1 unit 0.5

-2 -1 1 2

Arc Length Central Angle Formed Vertical distance Horizontal distance In radians In degrees 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Make a two scatterplots

If this triangle is a 30-60-90, what are the other two sides in terms of x (This is a special right triangle. If you don’t know google it.)

If this triangle is 45-45-90, what are the other two sides in terms of x

x What is the connection between degrees and radians?

Consider the following real numbers. Find the exact values (NOT a measured value, but a calculated value—no decimals) for these distances using what you know now based upon the discussion of the arc length, central angles, and radians for a unit circle.

Arc Central Angle Formed Vertical distance Horizontal distance length In radians In degrees From center From center 0  /6  /4  /3  /2 (2 )/3 (3 )/4 (5 )/6  (7 )/6 (5 )/4 (4 )/3 (3 )/2 (5 )/3 (7 )/4 (11 )/6 2

Questions

Why are all of these real numbers (arc lengths) based upon  ?

Do these values seem to correspond with the graph that you have constructing in the first portion? Explain the meaning of the sin( /2) as it relates to a triangle.

Why is the first value in the table labeled arc length and not angle? Could it be labeled as angles? For example: Is  /2 an angle or a length? Justify any response.

Is there a pattern that exists amongst your answers? Are any of the triangles congruent?

Why does the sine wave repeat? How long until it repeats? Consider the length required.