How to remember the special angles without the need to memorise or use the calculator

In math, there are group of angles known as special angles. They are 0, 30° (p /6 ), 45 ° ( p/4), 60° (p /3) & 90° (p/2) . These angles are usually needed in math topics such as trigonometry and complex numbers. Usually a calculator can be used however some answers require candidates to give it as the exact form meaning as a fraction. In this case there is no other way than to derive the sin, cos or tan of the special angles which often means memorising. However, there is a quicker way to derive it other than memorising as shown in the table below.

The key to using this type of table is simply the order in which for sinq is from left to right which is from 0 to p/2 radians with the exception that at 0 radians...the sin must be 0. From p/6 to p/2 radians onwards, the sin for that angle will be a fraction of denominator 2 with the numerator, the square root of 1 to 4 corresponding to the respective consecutive special angles. Obviously, some of the fractions can be simplified. Now once, the sin of the number can be derived, to find cos of the special angles is the opposite or reverse order from right to left. Whatever respective numbers that you have written for the sin q from left to right is now the cos q of the angles from right to left. For example, sin 0 = cos p/2 & sin p/6 = cos p/3 . Hence, it is very simple, to derive both sin & cos of the special angles. To find the tan of the special angles, you just have to know the formula that tan q = sinq / cosq and then you can use your already find the tangent based on the sin & cos of the corresponding angles that you have derived earlier. If you are used to the table, it can saved you time and as you get familiar with the table, you can sketch the table by around a min.

Does this table works for obtuse angles or larger angles?

As long as the angles can give basic angles which are the special angles, it will work. However, you must remember the quadrant rules,in which in the 1st quadrant all the Sin, tan & cos are positive, 2nd quadrant where only the sin is positive, the 3rd quadrant where only tan is positive and the 4th quadrant where only the cos is positive. I will illustrate with an example, 135° , the basic angle here is 45° and this angle lies in the 2nd quadrant and it happens that the basic angle is also a special angle. If you want to find the cosine it will be a negative since only sin is positive in this quadrant. So, based on the table, we find the cosine corresponds to 45° in the table and then add a - in front due to quadrant rule explained earlier on. This will give you the correct exact cosine of 135°. It also works for any other angles in the other quadrants as long as the basic angles are the special angles.

This table is quite useful as you can also use it for the inverse of sin, cos & tan to find the specific special angles. So really, it is useful if you guys could master this table.