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Problem 73

Think About It Sketch the graphs of $$f(x)=\sin …

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Problem 72

HOW DO YOU SEE IT? Consider an angle in standard position with $r=12$ centimeters, as shown in the figure. Describe the changes in the values of $x, y, \sin \theta, \cos \theta,$ and $\tan \theta$ as $\theta$ increases continually from $0^{\circ}$ to $90^{\circ} .$

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to asks us to consider an angle and standard position with a radius of 12 centimeters, as shown in this figure below. And with that, we're supposed to describe the changes in the values of X. Why? Scientist Tako San Data and Tangent data as data increases from zero all the way up to 90 degrees. So let's start by looking at the change in the behaviour of X. So when theta equals zero degrees, we can see that X equals 12 centimeters. All right, so that's when X is gonna beat its longest. Because we know that as ah Fada increases and approaches 90 degrees. This dog right here is going to trace the path of the unit circle or a circle with a radius of 12 centimeters. So just for visuals are drawn a rough circle. And we can see that as this point traverse is, you know, this ark and comes up here to why, at 90 degrees that are X value is constantly decreasing. Does it reaches 90 degrees X is gonna be zero. So the behavior of X is a decrease and decreases on the range from 12 20 All right, now, let's consider Why? Well, why is gonna have the opposite behavior when theta equals zero degrees? Why is going to be effectively equal to zero? But as we Trevor's, you know the path of that arc that has that 12 centimeter radius we see that why is actually growing. This pink dotted line right here will grow as our dot traces this line. And at 90 degrees it's going to be the same length as the Radius, just 12 centimeters. So why behaves in the opposite manner? It increases from zero to 12 now. What about science data? Well, let's think about what we know about circles in the unit circle and at zero degrees Sign is equivalent 20 Okay, so it's going to go from zero all the way up to whatever its value is at 90. Sign of 90 degrees is one, so we know that it's going to increase from 0 to 1 as our point traverses that 90 degree range. All right, so scientific. That will increase zero toe one. Now, what about co sign? Well, co sign is gonna do the opposite when we have zero degrees Coastline of zero. I'll just write these key values out, just so we know. Sign of zero equals zero sign of 90 degrees equals. One co sign of zero degrees equals one coastline of 90 degrees equal zero. And so we see the coastline is going to Trevor's from 1 to 0 over this 90 degree, you know, um, increasing our angle. Which means that co sign is going to decrease from 1 to 0. Now, what about tangent? Wrong tangent data. It's the same thing. Is signed data over Coast, same thing. And we know from our knowledge of rational expressions that we can never divide by zero, which means that when we reach 90 degrees, we're co sign of 90 degrees equal zero tangent data is going to be undefined. Okay, But since we know that science ADA is growing over this range and CO Santa is decreasing, tangent data must also be increasing because you have an increasingly larger number divided by an increasingly smaller number. And so when something big is divided by something small, we get something even larger. Okay, so tangent data is increasing from zero to infinity, and, um, it's undefined at 90 degrees because you can't divide by zero. But as it gets ever closer. That 89.9999999 degrees. We're going to find that tangent data gets closer and closer and closer to infinity.

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