Without using a calculator, determine which of the two values ...

Question

Without using a calculator, determine which of the two values is greater.

tan 1 or tan 2

Accepted Answer

Welcome back, everyone in this problem, we want to determine which of the following given trigonometric functions has a greater value. And we cannot use a calculator. We're deciding between a, the tangent of 1.3 and B the tangent of 2.3. Now, if we can't put this expression into our calculator, how are we supposed to figure it out? Well, the unit circle can help us because on the unit circle, if we get an idea of where these tangents will be, we'll know which ones will be positive or negative or which ones will be less or greater in comparison to the range for our quadrant. And then we should be able to solve. So let's first start by using our unit circle. So recall that on the unit circle, OK, on the unit circle, and I'm going to draw our unit circuit to the left here, gonna draw a good size circle for us to see what's really going on. And we know that on our unit circle, we have our X and Y axis. OK. And for our X and y axis, we also know that we have our quadrants for each part of the circle. We have our first quadrant and our first quadrant in terms of radiance is between zero is between zero and half of Pyres. OK. Our second project is between half of Pyridine and Pyridine or third quadrant. OK is between P and three halves of py radiance. OK? And then our fourth quadrant is between three houses of pi radians and two pi radians. I'm gonna write that below zero in blue just to represent that we make a full revolution comes around full circle. Now, what else do we know about our unit circle? Well, we know that the X and Y coordinates of our unit circle are in the form of the cosine of theta where the cosine is the X value or the X coordinate and the sine of theta where the sine of theta is the Y coordinate. Now, if we're trying to figure out or compare the tangent values, we need to understand how the tangent operates on our unit circuit. Do we know how our tangent relates to the cosine and sine function? Well, we also can recall that the tangent function is actually defined as the ratio of the sine function to the cosine function. So now we know how the tangent function works. We can put it on our unit circuit. Let's first start with a tangent of 1.3 and note that both of these are in radiance. OK. Now, if we're talking about the tangent of 1.3 where would 1.3 fall in our diagram? Well, a half of pi is approximately 1.6. So that means 1.3 would fall, the tangent of 1.3 would fall in our first quadrant. So we can represent it here. OK. Now what do we know about our tangent value in this quadrant? Well, here notice that in this quadrant, the X value is positive, the Y value is positive. So then that tells us that the tangent of 1.3 is going to be positive. OK, tangent of 1.3 is positive because the sine of 1.3 is positive and the cosine of 1.3 is positive. Next, let's think about the tangent of 2.3. Now 2.3 is is greater than a half of Pi which is approximately 1.6 but it's also less than pi which is approximately 3.14. So that means it's going to fall in our second quadrant. So let's represent that here. So if you put that in red, in red, that would actually be 2.3 radiance. Now, what do we know about the tangent in the second quadrant? Well, now in the second quadrant, our X value is negative and our Y value is positive, that means we'll be dividing a positive value by a negative value to get the tangent uh of 2.3. And when we divide a positive by a negative or a negative by a positive, vice versa. Then our result is negative. So that tells us then that the tangent of 2.3 is going to be negative. So now we understand the nature of both these tangents a bit, we can figure out which one has a greater value. It would have to be the tangent of 1.3 because it's positive. Therefore, our answer is answer choice. A thanks a lot for watching everyone. I hope this video helped.

Without using a calculator, determine which of the two values is greater.
tan 1 or tan 2

Key Concepts

Understanding the Tangent Function

Monotonicity of Tangent on Specific Intervals

Radian Measure and Interval Placement

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