Two Useful Trigonometric Tricks

One comment

Remembering trigonometric values for common angles is a major problem. Here is a trick that can ease this task. It was told by our mathematics instructor in Matric (class X). He told that you will not find it in any book. He was right. I have not found it in any book. It is so useful and easy to remember. It is given below.

For example the value of \sin(45^\circ) or \sin(\pi/4) will be \sqrt{2/4} which is equivalent to 1/\sqrt{2} or \sqrt{2}/2. Similarly the value of \cos(30^\circ) will be \sqrt{3}/2. For the values of \tan ignore the denominator and divide the corresponding value of \sin by the corresponding value of \cos. For example to compute the value of \tan(60^\circ), there is 3 written under \sin(60^\circ) and 1 written under \cos(60^\circ). Therefore the value of \tan(60^\circ) will be \sqrt{3}. Similar reasoning can be applied to calculate the values for \cot. The values for \sec and \csc can be calculated using their reciprocal relations.

 The second formula is called reduction formula that I found in a mathematical handbook. It is given by

f(\pm\alpha+n90^\circ)=\pm g(\alpha)
Or in terms of radian measure
f(\pm\alpha+n\frac{\pi}{2})=\pm g(\alpha)

Where,

n= any integer, positive, negative or zero.
f= any one of six trigonometric functions.
\alpha = any real angle measure.

(i)   If n is even, then g is the same function as f.
(ii)  If n is odd, then g is the co-function of f. \sin and \cos, \tan and \cot, and \sec and \csc are co-functions of each other.

The second \pm is determined by the quadrant in which angle (\pm\alpha+n90^\circ) or (\pm\alpha+n\pi/2) lies.

For example to calculate value of \tan(\alpha + 270^\circ), we note that 90^\circ is multiplied by 3 which is odd. Therefore g will be \cot. The corresponding angle lies in 4th quadrant where the value of \tan is negative. Thus we conclude that \tan(\alpha + 270^\circ) = -\cot(\alpha).

A pdf version of this post can be downloaded from here.

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1 comments on “Two Useful Trigonometric Tricks”

  1. This is an interesting way of remember trigonometry values. I have never seen it before, and can see how it works. However, I still prefer using the special triangles to recall these trig values. If you are not familiar with this method, I have posted about it on my site.

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