What Are Sine, Cosine, and Tangent Do you know what two angles living inside the same right triangle said to each other The first angle goes, Hey Thelma or is it Theta, I dont mean to go off on a tangent here, but whats your sine To which the second angle replies, Phil or is it Phi, I dont know why you even bother to ask, my sine is obviously the same as your cosine. Sine, Cosine and Tangent. And Sine, Cosine and Tangent are the three main functions in trigonometry. They are often shortened to sin, cos and tan. The calculation is. Lists the basic trigonometric identities. What do sines, cosines, and tangents have to do with right triangles And what are the sin, cos, and tan buttons on your calculator for Keep reading to find outSine, cosine, and tangent. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The word comes from the Latin sinus. Okay, so maybe thats not the best joke in the world, but once you understand sines and cosines, it is kind of funny. Of course, that means that if you dont know the difference between a sine and a cosine, youre currently left out in the metaphorical cold. Clearly we cant let that happenand we wont Because today were going to learn all about sines, cosines, and tangents. Recap Trigonometry and Triangles. When we talked about the world of trigonometry, we learned that the part of math called trigonometry is the part of math that deals with triangles. And, in particular, its the part of math that deals with figuring out the relationship between the three sides and the three angles that make up every triangle. Sin Cos Tan' title='Sin Cos Tan' />Of particular interest to us is the special type of triangles known as right triangles. Every right triangle has one 9. For our discussion of sine, cosine, and tangent which, dont worry, are not as complicated as they sound, its important that we have a way of labeling the sides of right triangles. As we learned last time, the longest side of a triangle is known as its hypotenuse. The side opposite the angle were looking at is known as the opposite side logically. And the side adjacent to the angle were looking at the one that isnt the hypotenuse is known as the adjacent side. Windows Longhorn Sound Scheme more. Sine, Cosine, and Tangent. With all of these preliminaries now happily splashing around inside our growing pool of mathematical knowledge, were finally ready to tackle the meaning of sine, cosine, and tangent. Heres the key idea. The ratios of the sides of a right triangle are completely determined by its angles. The ratios of the sides of a right triangle are completely determined by its angles. In other words, the value you get when you divide the lengths of any two sides of a right triangle lets say the length of the side opposite one of its angles divided by its hypotenuse is entirely set in stone as soon as the angles are set in stone. Why Well, if the angles are fixed, making the triangle bigger or smaller has no impact on the relative lengths of its sides. But changing the triangles angles, even a tiny bit, does If you need some convincing, try drawing a few triangles of your own and youll see that it is indeed true. Now, the fact that there are three sides of a triangle means that there are also three possible ratios of the lengths of a triangles sides. And, as you might have already guessed, these three ratios are none other than the famous sine, cosine, and tangent trigonometric functions. What Is SOH CAH TOA The sine of one of the angles of a right triangle often abbreviated sin is the ratio of the length of the side of the triangle opposite the angle to the length of the triangles hypotenuse. The cosine often abbreviated cos is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. And the tangent often abbreviated tan is the ratio of the length of the side opposite the angle to the length of the side adjacent. Since this is kind of a mouthful and a little hard to remember, kind folks over the centuries have come up with a handy mnemonic to help you and countless generations of kids in school out. All you have to remember is SOH CAH TOA. In other words. SOH  sin opposite hypotenuseCAH  cos adjacent hypotenuseTOA  tan opposite adjacentReal World Trigonometry. While all of this talk about the angles and sides of right triangles and their correspondence to one another through the beauty and magnificence of trigonometry is indeed lovely, it might leave you wondering a bit about the Why What and When of it all. By which I mean. Why exactly is this useful in the real worldWhat are the sin, cos, and tan buttons on my calculator for And how do they workWhen might I ever actually want to calculate the sine or cosine something Those, obviously, are all very important and very reasonable questions to ask. And theyre also very important questions to answer. Which is precisely the task we will begin to undertake next time. Wrap Up. Okay, thats all the math we have time for today. Please be sure to check out my book The Math Dudes Quick and Dirty Guide to Algebra. And remember to become a fan of the Math Dude on Facebook where youll find lots of great math posted throughout the week. If youre on Twitter, please follow me there, too. Until next time, this is Jason Marshall with The Math Dudes Quick and Dirty Tips to Make Math Easier. Basic Structural Engineering By Krishna Raju. Thanks for reading, math fans Trigonometry image from Shutterstock.