As we learned last time , the longest side of a triangle is known as its "hypotenuse. And the side adjacent to the angle we're looking at the one that isn't the hypotenuse is known as the "adjacent" side. Sine, cosine, and tangent With all of these preliminaries now happily splashing around inside our growing pool of mathematical knowledge, we're finally ready to tackle the meaning of sine, cosine, and tangent.
Here's 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—let's 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.
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 triangle's angles, even a tiny bit, does! If you need some convincing, try drawing a few triangles of your own and you'll 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 triangle's sides.
And, as you might have already guessed, these three ratios are none other than the famous sine, cosine, and tangent trigonometric functions. 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 triangle's 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. How will you use sine, cosine, and tangent outside the classroom, and why is it relevant?
There are a few career paths that lead to constant use of these equations. For example, lets say you're a sound engineer working on the production of a hit artist's new album. You know that sound travels in waves, and engineers can manipulate these waves measured by and applying trigonometry to create different computer-generated sounds.
What if you're an architect who needs to know the height of an existing building in a neighborhood you're assigned? You can use the distance you are from the building and the angle of elevation to detemine the height. You can even use trig to figure out the angles the sun will shine into a building or room. Construction workers also use sine, cosine, and tangent in this way.
They need to measure the sizes of lots, roof angles, heights of walls and widths of flooring, and even more. You can also see Graphs of Sine, Cosine and Tangent. To complete the picture, there are 3 other functions where we divide one side by another, but they are not so commonly used. They are equal to 1 divided by cos , 1 divided by sin , and 1 divided by tan :. Adjacent is always next to the angle And Opposite is opposite the angle.
How to remember? Think "Sohcahtoa"! The use of these trigonometric sin and cos has been rapidly increased in resolving engineering, navigation and physics problems. Sine is the first trigonometric function. For a right angle triangle, sine of an angle is the ratio of length of perpendicular or opposite side to hypotenuse.
In terms of expression. Trigonometric sine is used in studying the periodic phenomena of sound and light waves, determining the average temperature variations during the whole year, calculating day length, position of harmonic oscillators and many more.
0コメント