## How do the trig graphs relate to the Unit Circle?

When you straighten out the unit circle, it is easier to decipher how it is related to trig graphs. With our former knowledge of trig functions, we know that each trig function has its positive and negative values, depending on which quadrant it is located in. According to ASTC, we know that sine is positive in the first and second quadrant and negative in the third and fourth(+ + - -). We can take what we know from the unit circle and use it to help graph our trig functions by even using the points on the unit circle ( 0, pi/2, pi, 3pi/2, 2pi). Because sin is positive in the first and second quadrant, this part of the graph will be above the x-axis (the distance from 0-pi). Likewise, the part of the unit circle in the third and fourth quadrant will be in the negatives, below the x-axis(the distance from pi-2pi). Similarly to the unit circle, trig graphs can also be divided into "quadrants". For Cosine, it is positive in the first and fourth quadrant and negative in the second and third quadrant(+ - - +). As a result, when a cosine graph is plotted, it will be above the x-axis for the first quadrant, below the x-axis for quadrant 2 and 3 because it is negative. Then it rises above the x-axis again when it enters quadrant 4. for Tangent/Cotangent, it is positive in the 1st, negative in the 2nd, positive in the 3rd, and negative in the 4th. The pattern is basically + - + -. So when graphing , quadrant 1 and 3 will be above the x-axis while 2 and 4 will be below.

Please Refer to this picture as you read through the text to help clarify on what I mean by the Quadrants and how it ultimately affects the graph. Additionally, a sketch of a sin, cos, and tan graph are provided in relation to the quadrants.

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**a)** Period?

To understand why the period for sin and cos is 2pi, we must define a period. A period is one time through the cycle, one repeat of the pattern as you can say. When looking at a sin graph, we know that it is above the x-axis for the first 2 quadrants (from 0 to pi) and below for the last 2 quadrants (from pi to 2 pi). Thus the pattern is pos-pos-neg-neg, from 0 to 2pi. As a result, one repetition requires 2pi. Similarly, the pattern for cosine can be described as beginning positive from 0, once it reaches pi/2 the graph remains negative until reaching 3pi/2. From there it enters the fourth quadrant in which it is positive to the point 2pi. As a result the pattern can be simplified to pos-neg-neg-pos.This means that cos also requires 2pi for a period. However, tangent shifts from being positive beginning at 0 - pi/2. then it becomes negative from pi/2 - pi. From there, it becomes positive once again till it reaches 3pi/2. Then it becomes negative till 2pi. Thus the pattern is pos-neg-pos-neg. However, we see that the pattern repeats itself twice within one unit circle. Thus, one period will only need to be half of the unit circle which is pi. ### b) Amplitude?

Sine and Cosine have amplitudes because they are restricted to values ranging from -1 to1. We know that sin is y/r and cos is x/r. R can only be one because the radius of a unit circle is always 1. Because the radius is 1, the y and x values can not go beyond 1 meaning it has to be equal to or less than 1 or negative 1. When dividing the max value for y , which is one, by r, which has to be one, the greatest value we can have is 1. For these reasons, there are asymptotes, restricting the functions. As for the other trig functions, there are no amplitudes because there are no restrictions based on their ratios. For example, csc is 1/sin. The value of sine can range from -1 to 1. If the value was 0.1, then if we divide 1 by 0.1 we get 10. For these reasons, csc can't be restricted. The following applies to the rest of the trig functions. Tan is sin/cos, and the value of cos can range from 1 to -1. Thus we have an unrestricted value for tangent. This is the reasoning behind why**there are limits on sin and cos but no restrictions for the others.**

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