An Easy Way To Learn The Basic Trigonometric Identities

First learn the structure. Learn the positions of the six trigonometry functions. First comes the \sin function, underneath it comes the \cos function.

trigonometric identities

Then in the next column comes the \csc function, underneath it comes the \sec function. And in the last column comes the \cot function and underneath it comes the \tan function. Learning this position is important and just one time. The six trig functions are specially placed in the above given places, so as to serve our need.  Continue reading

The Trigonometric Ratio Hexagon

I found a book this morning containing a hexagonal figure of trigonometric ratios.  It reminded me of the mnemonics I created when I was still in high school in order to remember the formulas during exams. Unlike in other countries, we were not allowed to have formula sheets during exams, so we have to memorize them all.

Trigonometric Ratio

In the figure, any trigonometric ratio is a product of its immediate neighbors. Therefore,  Continue reading

The Complements Theorem

In this post, we discuss the proof behind one of the most commonly used identities in trigonometry. We examine the equations below and  show why the relationships always hold.

\sin \theta = \cos (90^{\circ} - \theta)
\cos \theta = \sin(90^{\circ} - \theta)

To students who have taken trigonometry, I’m sure that you have met these equation before.  The proof of these equations are as follows.

Consider triangle ABC right angled at C. From the definitions, we know that

\sin A = \displaystyle\frac{a}{c}
\cos A = \displaystyle\frac{b}{c}
\sin B = \displaystyle\frac{b}{c}
\cos B = \displaystyle\frac{a}{c}

 Therefore, (1) \sin A = \cos B and (2) \cos A = \sin B.

Now, If we let A = \theta, then B = 90^{\circ} - \theta, then substituting the values of A and B in (1) and (2), we have

\sin \theta = \cos (90^{\circ} - \theta)
\cos \theta = \sin (90^{\circ} - \theta)

 and these are what we want to show.

As exercises, use the strategy above, or any strategy you want to prove the following identities.

1.) \cot\theta = \tan (90^{\circ} - \theta)
2.) \sec \theta = \csc(90^{\circ} - \theta)