lol! it can be less than 1 too, upto -1! it cannot be greater than 1 because hypotenuse is always longer than the adjacent and opposite side... (from pythagoras theorem)
I've tried looking for the answer in Google, but I can't seem to understand it completely nor simplify the answers that I found. I was wondering if someone could explain this to me in much simpler terms. Thanks in advance guys :)
Regardless of the angle for which you evaluate tangent, cosine, or sine, you can always think of it as the ratio of two sides of a right triangle. See the orange angle (quadrant 1), green angle (quadrant 2), purple angle (quadrant 3), and red angle (quadrant 4) in the diagram.
For the tangent function, you can divide the opposite side by the adjacent side.
For the sine function, you can divide the opposite side by the hypotenuse.
For the cosine function, you can divide the adjacent side by the hypotenuse.
The story so far: Given one of the acute angles in a right triangle, you have studied two ratios involving the lengths of the sides of the triangle. The tangent ratio involved the length of the side opposite the angle divided by the length of the side adjacent to the angle. The sine ratio involved the length of the side opposite the angle divided by the length of the hypotenuse of the triangle. You have been playing favorites with the opposite side of the angle at the expense of the adjacent side. To even things up, let me introduce a new ratio, the cosine ratio. The cosine of an angle is the ratio of the length of the side adjacent to the angle divided by the length of the hypotenuse of the triangle. The cosine of A will be abbreviated as cos A.
You can play the same kinds of games that you played with the tangent and sine ratios.
- Example 4: If a right triangle has an angle with tangent ratio 9/14 , find the sineratio and the cosine ratio of the angle.
- Solution: Because a picture is worth a thousand words when working out these problems, I have sketched this situation in Figure 20.7. Because you need to know the sine and cosine ratios of the angle, you will need to calculate the length of the hypotenuse of the triangle. You can whip out the Pythagorean Theorem and take care of that right now:
- a2 + b2 = c2
- 92 + 142 = c2
- c2 = 277
- Now that you know the length of all three sides, finding the sine and cosine ratios can be done using the definition sin A = 9/277 and cos A = 14/277.
In a right triangle, the cosine of an angle is the ratio of the length of the side adjacent to the angle divided by the length of the hypotenuse of the triangle.
Figure 20.7A right triangle with tangent ratio 9/14.
At this point, you might be tempted to rationalize the denominator and write your answers as
- sin A = 9277/277 and cos A = 14277/277.
If so, then I would be impressed with your willingness to take things one step further, fearlessly going deeper into the torrential algebraic waters to write your answer in the form that I'm sure your algebra teacher emphasized when you first learned about radicals.
Of course, it is possible that the thought of rationalizing the denominator (as the process is officially called) never even occurred to you. That's okay, too. This isn't an algebra book, and there are advantages to leaving things in such a technically improper form. (Though when I wear my algebraic hat you'll never hear—or read—me say—or write—that it's okay to leave things improperly.)
One advantage to leaving things improperly is that you can clearly see that the sine and cosine ratios are less than . That's a quick and easy check to see if your answers make sense. The sine and cosine ratios can be equal to 1 on special occasions, but the ratios will never be greater than 1. Remember that the tangent ratio has no such restriction. You already saw that the tangent ratio can be greater, less than, or equal to 1.
You can work out these calculations in a variety of directions. If you are given the sine, cosine, or tangent ratio of an angle, you can find the other two ratios after using the Pythagorean Theorem.
The sine and cosine ratios of an angle cannot be greater than 1. The tangent ratio has no such restriction.
Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.
To order this book direct from the publisher, visit the Penguin USA website or call 1-800-253-6476. You can also purchase this book at Amazon.com and Barnes & Noble.
My son stumbled on an amazing graph completely by accident the other day. He’s doing some work reviewing trig functions this week and I asked him to just play around with some graphs in Mathematica to get a feel for how Sin[x] and Cos[x] behave. One of the graphs he drew was:
from to :
I certainly wasn’t expecting him to make a graph like this one, but was happy that he did. Yesterday we talked through what was going on.
We started by discussing why the graph seemed so strange:
Now we dove into some of the details – which involve complex numbers and the definition:
as well as the definition of even and odd functions. So, there’s a lot of math to that we need to bring to the table to understand what’s going on in our graph.
Finally, we calculated the exact value of . Again, there’s a lot of advanced math that comes in to the calculation here – but even if some of the math ideas took a bit to sink in, I’d say that all in all it was a good conversation:
View all posts by mjlawler