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

Question

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

cos 2 or sin 2

Accepted Answer

Welcome back, everyone. In this problem, we want to determine which of the following given trigonometric functions has a greater value without using a calculator. We're comparing a to cosine of 1.4 to B, the sine of 1.4. Now, if we're not using a calculator, how are we supposed to figure out which one is greater? Well, we can use the unit circle to help us and there are a lot of different ways that we can think about. But let's start with the unit circle. Now we can use the unit circle. OK? Because on the unit circle, let me drive it on the left, on the unit circle, we can see a trigonometric function in all its quadrants. OK. So let's say that we have our X axis here and our Y axis vertically. And we know that our unit circle is going to be split up into the units from zero to half of pi which is quadrant one from half of pie to pie, which is quadrant two from pie to three halves of pie, which is quadrant three and from three halves of pie back to two. Pi let me write two pie here just to represent that it's a complete revolution which would have been our fourth quadrant. We also know that on our unit circle, the X and Y coordinates are actually the functions of the cosine function of the cosine of an angle. Sorry. And the sine of an angle where the cosine is the X coordinate and the sine function is the Y coordinate. So if we're representing or trying to figure out whether the cosine of 1.4 or the sine of 1.4 is greater, we can put it on our unit circle and then try to brainstorm from there. So let's go ahead and do that. Now, what do we know about 1.4 radiance? Because remember these are in units radiance. Well, 1.4 radiance would fall in the first quadrant because a half of pi is approximately 1.6. So 1.4 would fall somewhere there. OK. So let's say that that is our angle for 1.4 radiance. So that means our X value, let's put that in red. OK. Our X value cosine would be here. OK? So that's the cosine of 1.4. While our Y value here would have been the sign, the sign of 1.4. So now we know where they are. But the thing is because both of them fall in the same quadrants, it's still a little bit difficult to tell which one has a greater value. So what else do we know? What else do we know? Well, let's think about where in the first quadrant, both the cosine and sine functions would be equal. So that would make it a bit easier to compare. Now, recall, OK, recall that halfway between zero and half of pi which is 1/4 of pi the cosine function is equal to the sine function. OK. So this is, this is a given, this is something that we know right now. What does that tell us? Then 1/4 of Pi would probably land somewhere right here. 1/4 of Pi would probably land somewhere right here. OK. Fourth of Pian and four or fourth of Pi Radians, it would be approximately 0.8. So that means 1.4 would be greater than 1/4 of Pi. Now, at that point on our diagram, if the cosine of 1/4 of pi equals the sine of 1/4 of pi, notice that above 1/4 of Pi in that region where the sine and cosine of 1.4 are in this region. OK. Notice that in this region, the sine values, if we're moving from 1/4 of pi, the sine values start to get larger. OK. While the cosine values, the X values start to get less. So that means at this point at the point we're interested in most likely or definitely rather the sine value would be greater than the cosine value because they were getting larger while the sine values were getting smaller. So what does that tell us then? Well, based on our diagram, based on our, our logic, that must mean that the sine of 1.4 is going to be larger than the cosine of 1.4. If we look back on our answer choices, that would have been 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.
cos 2 or sin 2

Key Concepts

Understanding the Range and Behavior of Sine and Cosine Functions

Evaluating Trigonometric Functions at Specific Angles Without a Calculator

Comparing Numerical Values of Trigonometric Functions

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