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.
You will be expected to use these values to provide "exact" answers for solving right triangles and for finding the values of various trigonometric ratios.
Usually, textbooks present these values in a table that you are expected memorize. But pictures are often easier to recall on tests, etc, at least for some of us. If those tables aren't working for you, then this lesson will show the way in which many people (including me!) really keep track of these values.
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In what follows, I use degrees for the angle measures. This is usually how students are introduced to angle measures. However, in case you're working with radians, I'll also note the radian anglemeasure equivalents.
All 454590 triangles are similar; that is, they all have their corresponding sides in ratio. (An angle measuring 45° is, in radians, .) So let's look at a very simple 454590:
The hypotenuse of this triangle, shown above as 2, is found by applying the Pythagorean Theorem to the right triangle with sides having length . The base angle, at the lower left, is indicated by the "theta" symbol (θ, THAYtuh), and is equal to 45°. So how does knowing this triangle help us?
It helps us because all 454590 triangles are similar. Therefore, every "evaluation" or "solve the triangle" question involving a 454590 triangle or just a 45° angle can be completed by using this triangle. This picture is all you'll need.
When we need to work with a 30 or a 60degree angle, the process is similar to the above, but the setup is a bit longer. (A 30° angle is equivalent to an angle of radians; a 60° angle is equivalent to an angle of radians.)
For either of the angles, this is the triangle that we start with:
This is a 606060 triangle (that is, an equilateral triangle), with sides having a length of two units.
We drop the vertical bisector from the top angle down to the bottom side:
Note that this bisector is also the altitude (height) of the triangle.
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By using the Pythagorean Theorem, we get that the length of the bisector is . And this bisector has formed two 306090 triangles.
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When we are working with a 60degree angle, we use the lefthand triangle above, at it stands, with the base angle (at the left) labelled "α" (ALphuh, being the funnylooking "a"):
When we are working with a 30degree angle, we use the righthand triangle, knocked over to the left, base angle (at the left) labelled "β" (BAYtuh, being the funnylooking "b"):
We can find trigonometric values and ratios with the 30degree and 60degree triangles in the exact same manner as with the 45degree triangle. The above pictures are all you'll need.
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.
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The pictures above are what I've always used, and many find them helpful. On the other hand, some people prefer tables or other methods. If tables work better for you, then this table comes highly recommended, having been "fieldtested" by a working instructor:


30° 
45° 
60° 
sin 
1 
2 
3 

cos 
3 
2 
1 



divided by 2 
To find, say, the sine of an angle measuring 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 . 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, . 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°.
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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 square root of the number of fingers to the left by the square root of the number to the right (and rationalize, as necessary).
For instance, if you're wanting to work with a thirtydegree angle, you'd orient your hand like this:
The sine is the square root of your thumb (that is, the square root of "one") over two, which yields:
The cosine is the square root of your three fingers (that is, the square root of "three") over two, yielding:
On the other hand, if you're wanting to evaluate sin(0°), cos(0°), and cot(0°), you'd orient your left hand like this:
Since your thumb is folded down, there are 0 fingers to the left, and 4 fingers to the right. Then the values of the sine and the cosine are found as:
The cotangent is the reciprocal of the tangent. What is the value of the tangent?
Flipping the above would cause division by zero, which isn't allowed. So cot(0°) is undefined.
(An angle measuring 0° is equivalent to an angle of 0 radians. An angle measuring 90° is equivalent to an angle of radians.)
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