Trigonometry involves calculating angles and functions of angles, such as the sine, cosine and tangent. Calculators can be handy in finding these functions because they have sin, cos and tan buttons. However, sometimes you won't be allowed to use a calculator on a homework or exam problem or you might simply not have a calculator. Don't panic! People were calculating trig functions long before calculators came along, and with a few simple tricks, so can you.
Trig Functions of Graphical Axes
The axes on a standard graph are at 0 degrees, 90 degrees, 180 degrees and 270 degrees. It is simplest to memorize sine and cosine functions for these special angles because they follow easy-to-remember patterns. The cosine of 0 degrees is 1, the cosine of 90 degrees is 0, the cosine of 180 degrees is –1, and the cosine of 270 is 0. Sine follows a similar cycle, but it begins with 0. So the sine of 0 degrees is 0, the sine of 90 degrees is 1, the sine of 180 degrees is 0, and the sine of 270 degrees is –1.
Often when you are asked to calculate the trig function of an angle without a calculator, you will be given a right triangle, and the angle you are asked about is one of the angles in the triangle. To solve these types of problems, you need to remember the acronym SOHCAHTOA. The first three letters tell you how to find the sine (S) of an angle: the length of the opposite (O) side divided by the length of the hypotenuse (H). For example, if you are given a triangle whose angles are 90 degrees, 12 degrees and 78 degrees, the hypotenuse (the side opposite the 90-degree angle) is 24, and the side opposite the 12-degree angle is 5. You would therefore divide the opposite side by the hypotenuse, 5/24, to get 0.21 as the sine of 12 degrees. The remaining side is called the adjacent side, and it is used to calculate the cosine. The middle three letters in SOHCAHTOA indicate that the cosine (C) is the adjacent side (A) divided by the hypotenuse (H). The final three letters tell you that the tangent (T) of an angle is the opposite side (O) divided by the hypotenuse (H).
The 30-60-90 and 45-45-90 triangles are used to help remember trig functions of certain commonly used angles. For a 30-60-90 triangle, draw a right triangle whose other two angles are approximately 30 degrees and 60 degrees. The sides are 1, 2 and the square root of 3. The smallest side (1) is opposite the smallest angle (30 degrees). The largest side (2) is the hypotenuse and is opposite the largest angle (90 degrees). The square root of 3 is opposite the remaining 60-degree angle. In the 45-45-90 triangle, draw a right triangle whose other two angles are equal. The hypotenuse is the square root of 2, and the other two sides are 1. So if you are asked to find the cosine of 60 degrees, you would draw the 30-60-90 triangle and notice that the adjacent side is 1 and the hypotenuse is 2. Therefore, the cosine of 60 degrees is 1/2.
If you are not given a triangle or a special angle, you can resort to using a trig table, in which certain trig functions have been calculated and tabulated for each degree between 0 and 90. An example trig table is provided in the Resources section of this article.