A representation of what [math]\pi[/math] is, its relation to the unit of angle measure called radians, and the difference between radians and degrees.[br][br]In this worksheet, the gridlines are separated by r units, where r is the radius of the two circles. d represents the distance the center of the circle has moved. As you change d by dragging the slider or by directly inputting the value at the bottom left, the blue segment gets wrapped around the circle. (Note: d also has units of r.)

[b]1. [math]\pi[/math] is defined to be the ratio of the circumference of a circle to its diameter.[/b][br]How do we know its value? Go through this problem to see one way of approximating it.[br][br](A) When d=[i]x[/i], the distance the center of the circle has moved should be ___.[br](B) When d=[i]x[/i], the length of the portion wrapped around the circle should be ___. Check your answer with the number in blue.[br](C) What are (A) and (B) in terms of the diameter? Hint: d is in units of radiuses and radius = 1/2*diameter.[br](D) When [math]d \approx 3.141592[/math], how much of the circle appears to be wrapped?[br](E) Remember d=3.141592 means the center has moved 3.141592 radiuses over. What is (D) in terms of the diameter?[br](F) Repeat (D) and (E) for [math]d \approx 6.283184[/math].[br](G) [math]\pi[/math] = (circumference)/(diameter) = # of times a diameter can be wrapped around its circle. So based on (F), [math]\pi \approx[/math]___.[br][br]If you haven't already done so, slide d from 0 all the way to [math]2\pi[/math] to see what you just did, without interruptions. Note that [math]\pi \neq 3.141592[/math]. Instead, [math]\pi \approx 3.141592[/math]. [math]\pi[/math] is actually an irrational number.[br][br][br][b]2. Radians are a unit of [math]\angle[/math] measure defined to be the # of radiuses that go into the arc subtending the [math]\angle[/math].[/b][br]This makes sense since the angle is completely determined by how long the arc is [i]compared[/i] to its radius.[br][br](A) When d=1, the arc wrapped around the circle has length ___. Check your answer with the number in blue.[br](B) How many radiuses can be wrapped around the arc from (A)? Hint: d is in units of radiuses.[br](C) [math]m\angle[/math] subtended by the arc from (A) (i.e. the green [math]\angle[/math]) should be ___ radians. Check your answer with the value in green.[br](D) Repeat (A)-(C) for d=x.[br][br]Notice that arc length and radius both have units of length. Since radians = (arc length)/(radius), you can see that radians are dimensionless (i.e. they are pure numbers). Also notice that [math]2\pi[/math] radians = 360[math]^{\circ}[/math]. So to convert from radians to degrees, multiply by [math]180^{\circ}/\pi[/math].[br][br]Radians are a much more natural unit of [math]\angle[/math] measure than degrees, and therefore, most of mathematics uses radians instead of degrees. However, in this course, you will not be using radians. [b]All [math]\angle[/math] measures will be given in degrees[/b]. Therefore, when doing trigonometry problems, [b][i]MAKE SURE YOUR CALCULATOR IS SET TO DEGREES, NOT RADIANS[/i][/b]. Typically, you can see which mode you're in by looking for a "deg" symbol or a "rad" symbol on the screen. If you're in the wrong mode, change it, or else your answers for trigonometry problems will likely come out all wrong.