Some angles occur more frequently in trigonometry than the others. Such angles measure 30 degrees, 45 degrees and 60 degree and are located in the first quadrant of a rectangular coordinate system. You may notice that these angles are associated with the special right triangles 45-45-90 and 30-60-90. How do we find the trigonometric functions of these angles? Once we know these, we can get the values for all these angles in the second, third and fourth quadrants as well. This knowledge will be very helpful in solving many problems in trigonometry.
Recall in geometry that a 45-45-90 right triangle with two 45 degree angles and a 90 degree angle Assume that each leg of the triangle has length 1. Therefore, by the Pythagorean Theorem, the hypotenuse is √2. The trigonometric functions sine, cosine and tangent are then 1/√2, 1/√2 and 1, respectively. Note that the values of sine and cosine of 45 degrees is generally noted as √2/2.
Note that a 45 degree angle in the second quadrant is 180 – 45 = 135. A 45 degree angle in the third quadrant is 180 + 45 = 225 and a 45 degree angle in the fourth quadrant is 360 – 45 = 315. The values are the same in each quadrant with the exception of their signs. In the second quadrant sine is positive, cosine is negative and tangent is negative, so the values would be √2/2, -√2/2 and -1, respectively. In the third quadrant, sine and cosine are both negative and tangent is positive. In the fourth quadrant sine is negative, cosine is positive and tangent is negative.
Now, recall in geometry that a 30-60-90 right triangle is a triangle with a 30 degree angle, a 60 degree angle and a 90-degree angle. Assume the hypotenuse to have length 2. The side opposite the 30 degree angle is half the length of the hypotenuse, so it’s equal to 1. The side opposite the 60 degree angle is always √3 times the shortest leg, so it’s √3. Therefore, the sine of 30 degrees which is opposite divided by hypotenuse is 1/2. The cosine of 30 degrees is the adjacent side divided by the hypotenuse which is √3/2 and the tangent is sine divided by cosine, which is 1/√3 or √3/3. Remember the signs of these values change depending on the quadrant the angle is located. In the second quadrant the angle is 150 degrees, in the third it’s 210 degrees and in the fourth it’s 330.
For the 60 degree angle, the length of the side opposite is √3, the length of the side adjacent is 1 and the length of the hypotenuse is 2. Therefore the sine, cosine and tangent of 60 is √3/2, 1/2 and √3, respectively. A 60 degree angle in the second quadrant is 120 degrees, in the third quadrant it’s 240 and in the fourth quadrant it’s 300.
Notice the sine of 30 is the same as the cosine of 60, and vice versa. Sine and cosine of 45 are the same. There is an easy way to remember the values of sine and cosine of 30, 45 and 60. All have a denominator of 2 and the numerators for sine are 1, √2 and √3, and reverse order for cosine.
The values for sine and cosine are easy to remember for every 90 degrees as well. Starting at 0, sine is 0 and cosine is 1. For 90 degrees, the values are opposite, sine is 1 and cosine is 0. The values alternate again for 180 degrees expect they are negative so it’s 0 and -1. For 270 degrees they alternate again with the negative sign remaining, so it’s -1 and 0. So the values in (sine, cosine) form are (0, 1), (1, 0), (0, -1) and (-1, 0).
Learning the values of sine and cosine for these special angles from 0 to 360 degrees will greatly help solve many problems in trigonometry.