Unit Circle and Trig Function GSP Tour -Open the UnitCircleTrig file posted on the website and explore it to fill in the following blanks. On a clean sheet of notebook paper, you must write this out (no print outs) and underline, highlight or use a different color for the words that go in the blanks. Not doing so will result in no credit. When I open the gsp file I notice a circle centered on a graph. The x-axis has units of ________ and goes from about _______ to about ________. This is for plotting the trig functions ______, _______ and _______ because there are _______ degrees in a complete circle. The y-axis goes from about ______ to about _____ because it represents the radius of the unit circle which by default is _______ unit(s). The unit circle (and any circle) can be graphed by using either a regular function based on the _____________, such that the radius is equivalent to the _______ of the two sides squared. In the case of the unit circle the function would be y = ____________. To graph a circle with radius ‘r’, the function would change to y = ____________. A circle itself is not a true function because it does not pass the ____________ test or in other words there are multiple ____ values for each _____ value. In this case however we are taking the ________ root of (r2 + x2) to solve for ______ when we have y2. We know that anytime we take the square root of a number we wind up with _____ results, one is ______ and one is ________ because the square of a positive number as well as the negative of that number with both give us the same number (i.e. (3)2 = 9 and (-3)2 = 9 ). This means that the function for a circle is essentially two functions, one that has ____ values that are positive and one that has ‘y’ values that are _________. The ________ form of the circle equation is: y2 + x2 = r2. Adding to the _______ values would move the circle up while subtracting from the ______ values would move it down. Adding to the _____ values would move the circle to the right while subtracting from them would move it left. The trig functions can also be used to graph a circle by splitting the ‘x’ and ‘y’ values into ___________ equations (vector eq’s). The trig functions explain the lengths of the _________ of the triangles formed by the central angle of the circle: (the variable used for that angle is commonly _______ ). To find the leg that is the ‘x’ length we use the _______ function and to find the leg that is the ‘y’ length we use the ________ function. The conversion factor to change from degrees to radians is: 1 radian = ________ and to convert from radians to degrees (solve for degrees algebraically): 1 degree = _________. As we move around the circle we are at _______ radians when we are at 180 degrees. When we are at _______ radians we are at 53 degrees. When we are at 3.5π radians we are at _______ degrees and when we are at 0.3 π radians we are at _______ degrees. (click show sin then drag point „x‟ or click the „Move around the circle‟ button) Observing what happens with the triangle and the graph of sin of the angle theta, I notice that the values on the x-axis represent ________ and on the y-axis represent the _______ which is the ________ of the circle at 90 and 270 degrees. The sin function is plotted as we move along the x axis. If we start at 0 degrees the y value, which is the same as the ________ side of the triangle formed by angle theta, is equal to ________ . At 45 degrees (around the circle and on the x-axis, we have a value of ______ for y, which is the ________ of 45 degrees. As we move continuously around the circle, the value of ______ changes because it is equal to the sin of the angle ______ which is also the ________ side of the triangle formed by theta. (click hide sin and show cos buttons and use key-binding “Ctrl B” then drag point „x‟ or click the „Move around the circle‟ button) As we plot cos (as when we plotted sin), we move along the ______ axis which is the number of degrees for angle _______. As we move around the circle, a right triangle is created by the _______ and _______ axis along with a __________ line drawn from the x-axis. The cos of angle theta is equal to the ratio of the ________ side over the _______ or _________ of the circle. We can see that at 45 degrees with a radius of 1, this length is the ________ of a 45, ____, _____ special right triangle. That means that cos 45 = _______. As we move closer to 90 degree the size of this leg becomes ________ until it is essentially ________. Moving past 90 degrees toward 180 the leg then begins to lengthen in the _________ direction. It is at a max or min length at 0 degree and _______ degrees respectively. At these angles it is + or - ________ unit(s). (click hide cos and show tan buttons and use key-binding “Ctrl B” then drag point „x‟ or click the „Move around the circle‟ button) The tangent function looks _________ than the sin or cos functions in that it moves up toward _________ or down toward _________________. The tangent of angle theta is the ratio of the ________ side to the _________ side. This means that the numerator is the ________ side and the denominator is the ________ side. This being the case, as we move from 0 degree up to 90, the numerator gets __________ while the denominator gets _________. When this happens the resulting number approaches __________. When we move from 90 to 180 degrees the opposite begins to happen and the numerator gets ____________ while the denominator gets _____________. The sine and cosine function both have ______ and ______ values for their ranges. These values create what is known as the amplitude of the cyclical function. In the case of the unit circle it is from ________ to _______ whereas for any circle of radius ‘r’ it would be from ______ to _____. If we travel from 0 to 360 degrees around the circle we complete on full cycle. This is known as the wavelength represented by the Greek letter lambda (λ). In terms of degrees or radians the wavelength is also considered one full cycle and will be the _________ for all circles. From this concept we derive frequency which is simply the number of ________ or ________ per unit of time. Amplitude yields intensity while frequency gives us different musical notes and visual colors.