2. How do the trig graphs relate to the Unit Circle?
If we "unwrap" the unit circle and put it onto the horizontal axis of a standard graph, we see the values of the trig functions would correspond. For example, sine is only positive in the first and second quadrants of the unit circle, so when we unwrap the values of sin would be positive from 0 to 180 degrees or 0 to π. Then, after π, all the sine values will become negative until the graph reaches 2π and it becomes zero. Also, we know that sine of 0 degrees (1,0), 180 degrees (-1,0), and 360 degrees (1,0) is all equal to zero since it's just y/r. That is why sine graphs start and end at zero while being zero again in the middle.
In addition, cosine is only positive in the first and fourth quadrants of the unit circle. So from 0 to π/2, the values of cosine would be positive, become negative from π/2 to 3π/2, and become positive from 3π/2 to 2π. However, the graph starts and ends at 1 because the cosine of 0 and 360 degrees (1, 0) is equal to 1.
For tangent, it is only positive in the first and third quadrant of the unit circle. So from 0 to π/2 the values are positive, negative between π/2 to π, becomes positive again from π to 3π/2, and negative from 3π/2 to 2π. Or you could look at the trig ratios since tan= sin/cos. So whenever sine or cosine is negative, tangent will also be negative. Or if both functions are positive/negative, then tangent will be positive.We also have to make sure that we add the asymptotes, since tangent would become undefined whenever cosine is equal zero. Sine and cosine will never have asymptotes since their denominator of r will always be 1 and not undefined.
a) Period? Why is the period for the sine and cosine 2π, whereas the period for tangent and cotangent is π?
First, we have to know that the period of a trig function is defined as the length of the cycle of the curve.Sine and cosine will always have a period of 2π since it will take the complete distance of the unit circle for the graph to start repeating itself again. Sine is positive(1st quadrant), positive (2nd quadrant), negative (3rd quadrant), negative (4th quadrant), and continues the same thing over and over again. Cosine is positive (1st quadrant), negative (2nd quadrant), negative (3rd quadrant), positive (4th quadrant), and on and on and on. Tangent and cotangent only have a period of π because after π (180 degrees), the graph is already repeating itself again. Tangent and cotangent is positive (1st quadrant), negative (2nd quadrant), positive again (3rd quadrant), negative again (4th quadrant), etc.
b) Amplitude? How does the fact that sine and cosine have amplitudes of 1 (and the other trig functions don't have amplitudes) relate to what we know about the Unit Circle?
Amplitudes are half the distance between the highest and lowest points on the graph. Sine and cosine both have an amplitude of 1 because the Unit Circle is restricting their values. On the Unit Circle the highest and lowest values for sine and cosine would be –1
and +1. They could be anything from –1
and +1, but not anything higher or any lower because there were no other points beyond them. That is why we had no solution every time sine or cosine was equal to number greater than 1 or less than -1. Tangent and cotangent, on the other hand, never had any such restrictions since they were equal to the ratio of sine and cosine.Since tangent and cotangent go on forever (to positive and negative infinity), then they won't be able to have or find an amplitude. Its range will always be (negative infinity, positive infinity), making no definite highest and lowest point.Secant and cosecant will also have no amplitudes because they will also be under no restrictions from negative infinity to a certain point, and another certain point to positive infinity. Therefore, they too do not have definite highest and lowest points.