Sines - Triangles and Beyond!
A Guide to Sines
In trigonometry (literally - triangle measure) people study the properties of trigonometric functions like the sine, cosine and tangent. In a right triangle (that's a triangle where one angle is 90 degrees, or made from two perpendicular lines), the sine of an angle is the ratio of the side opposite the angle to the hypotenuse (the longest side). That is, to get the sine of an angle on a right triangle, measure the side opposite the angle, measure the hypotenuse and then divide the first by the second.
Of course, these days, to get a sine, you turn on a calculator and press the button for sine. Usually it's labeled sin. Take the sine of 58 degrees. It's 0.85. The sine of 90 degrees is 1, because the side opposite the right angle IS the hypotenuse, and the ratio of something to itself is 1. But if you enter 100 and push sine, you get 0.98. How can you have an angle of 100 degrees in a right triangle? You can't. The angles of the three sides of any triangle are 180 degrees. One angle of a right triangle is 90. 100 + 90 = 190 which means the other angle would have to be negative. A negative angle? What? Well, mathematicians don't let that stop them.
Just define sine a little more generally.
How do we (or they) define a sine?
Draw a circle that has a radius of 1 unit (it could be 1 inch, for example).
Draw an x and a y axis through the center of the circle.
Draw a line from the center to any point on the edge (the circumference). Measure the angle of that line to the horizontal axis (or x-axis). Call this angle theta (the Greek letter). Then draw a horizontal line from the point on the edge to the y-axis (the vertical one). The point where the line crosses the y-axis is the sine of theta.
Why would we do all this? Well, the sine has many useful properties. But some of them depend on taking the sine of angles greater than 90 degrees.
More about the uses of sines in another article.
Then the sine of theta
Source: http://mathworld.wolfram.com/Sine.html
Of course, these days, to get a sine, you turn on a calculator and press the button for sine. Usually it's labeled sin. Take the sine of 58 degrees. It's 0.85. The sine of 90 degrees is 1, because the side opposite the right angle IS the hypotenuse, and the ratio of something to itself is 1. But if you enter 100 and push sine, you get 0.98. How can you have an angle of 100 degrees in a right triangle? You can't. The angles of the three sides of any triangle are 180 degrees. One angle of a right triangle is 90. 100 + 90 = 190 which means the other angle would have to be negative. A negative angle? What? Well, mathematicians don't let that stop them.
Just define sine a little more generally.
How do we (or they) define a sine?
Draw a circle that has a radius of 1 unit (it could be 1 inch, for example).
Draw an x and a y axis through the center of the circle.
Draw a line from the center to any point on the edge (the circumference). Measure the angle of that line to the horizontal axis (or x-axis). Call this angle theta (the Greek letter). Then draw a horizontal line from the point on the edge to the y-axis (the vertical one). The point where the line crosses the y-axis is the sine of theta.
Why would we do all this? Well, the sine has many useful properties. But some of them depend on taking the sine of angles greater than 90 degrees.
More about the uses of sines in another article.
Then the sine of theta
Source: http://mathworld.wolfram.com/Sine.html
Published by Peter Flom
I am a statistician, working with a wide variety of clients, mostly researchers in psychology, education, medicine, social sciences and other fields. I also have given talks and written articles on learning... View profile
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2 Comments
Post a CommentDitto what Michele said! ;-)
Sine, cosine - I vaguely remember this :) cheers!