
Special Angle Values There are a few (a very few) angles that have relatively "neat" trigonometric values, involving, at worst, one square root. Because of their relatively simple values, these are the angles which will typically be used in math problems (in calculus, especially), and you will be expected to have these angles' values memorized. Usually, textbooks present these values in a table that you are expected memorize. But pictures are generally easier to recall (on tests, etc); this lesson will show the way in which many people really keep track of these values. 45°Angle Values (from a 454590 triangle)
We choose a length of one unit for the matching sides because this is simplest, and then we get the sqrt(2) value by using the Pythagorean Theorem. (Note that it is irrelevant what are the lengths of the actual triangle you are dealing with; this reference triangle gives you the ratios, and thus the trigonometric values.) The "theta" (the squashed circle with the line through it) in the lower left corner is the angle we'll use. (The orientation of the particular angle they want you to work with is irrelevant, because you can always rotate the above triangle to put it into whatever orientation you need.)
If, for a given exercise, they only want the trigonometric value of the angle, then just read it off the triangle: the sine of theta is (opposite) over (hypoteneuse), or 1/sqrt(2); the cosine of theta is (adjacent) over (hypoteneuse), or 1/sqrt(2); and the tangent of theta is (opposite) over (adjacent), or 1/1 = 1. (Your text or teacher might want you to "rationalize" these ratios, in which case you get sin(theta) = sqrt(2)/2 and cos(theta) = sqrt(2)/2. This rationalization issue becomes much less important once you reach calculus.) On the other hand, suppose they've given you a 45degree triangle where the two matching sides have length, say, 14 units, and they want you to find the length of the third side. Okay, you multiply 1 by 14 to get 14 for the matching sides. Then, by similar triangles, you multiply sqrt(2) by 14 to get 14sqrt(2) for the length of the hypoteneuse. Or suppose they tell you that the hypoteneuse is 12 units. Let's figure out what we have to multiply by to get the lengths of the matching sides. Let this multiplier be "x". Then xsqrt(2) = 12, so x = 12/sqrt(2) = 12sqrt(2)/2 = 6sqrt(2). Then the lengths of the matching sides are 1×6sqrt(2)= 6sqrt(2) units. Copyright © Elizabeth Stapel 20002011 All Rights Reserved You can find anything you need from this reference triangle. Instead of trying to memorize an entire table (if that's not working for you), simply memorize this triangle. 30° and 60°Angle Values (from a 306090 triangle) If you need to work with a 30 or a 60degree angle, the process is similar to the above, but the setup is a bit longer.
You find trigonometric values and ratios with the 30degree and 60degree triangles in the exact same manner as with the 45degree triangle. You may get one of those teachers who doesn't want you to draw these pictures (because you're supposed to have everything memorized by now). Well, this is why your pencil has an eraser. My Calculus II instructor said that if we drew the pictures on our tests, the entire problem would be counted wrong. I drew the pictures anyway, but very lightly, and erased them all before I handed the tests in. He never knew, and I passed the course. You do what you gotta do. On the other hand, some people prefer tables and charts. If tables work better for you, then this table comes highly recommended, having been "fieldtested" by a working instructor:
To find, say, the sine of fortyfive degrees, you would trace across in the "sin" row and down the "45°" column, taking the squareroot symbol with you as you go and remembering to include the "divided by 2" from the bottom, to get sin(45°) = sqrt(2)/2. The neat pattern of "1, 2, 3" across the top row and "3, 2, 1" across the middle row are meant to help you memorize the table values. Keep in mind that the square root of 1 is just 1, so, for instance, cos(60°) = sqrt(1)/2 = 1/2. To find the tangent, you'd divide the sine value by the cosine value. Another method uses your left hand to essentially do the same thing. With your palm facing you, count off the basic reference angles, starting with your thumb: 0°, 30°, 45°, 60°, and 90°. To find a trig value, you'll lower the finger corresponding to that angle, keeping your palm facing you. For the sine value, you'll take the square root of the number of fingers to the left of the lowered finger, and divide by 2; for the cosine value, you'll take the square root of the number of fingers to the right of the lowered finger, and divide by 2; for the tangent, you'll divide the number of fingers to the left by the number to the right and take the square root of that fraction.


This lesson may be printed out for your personal use.

Copyright © 20002014 Elizabeth Stapel  About  Terms of Use  Linking  Site Licensing 




