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Understanding Sine And Cosine

20 November 2014 - Berkeley, California

Do you want to understand what $sinx$ and $cosx$ are once and for all? Do you want to look at the formula $sin^2x + cos^2x=1$ and know exactly how to prove it? If the answer is yes, continue reading.

So, what is $sinx$? Let’s say we have a right triangle with the sides a, b, c and the angle x between c and b.


Then, by definition

That’s it! You take an angle in the right triangle, divide the opposite side by the longest side (the hypotenuse), and you get the sine of this angle. Note that $sinx$ does not depend on the size of a right triangle, as long as it contains the same angle $x$, since all such triangles are similar.

Cosine, tangent and cotangent are defined similarly:

These are just the short way to refer to the ratios of the sides of the right triangle. There is no magic.

Now, why is $tanx={sinx \over cosx}$?

Why is $tanx=\frac{1}{cotx}$?

Can you prove that $cotx=\frac{cosx}{sinx}$?

Everyone has seen this formula before

It is called Pythagorean trigonometric identity. Where this formula comes from is not usually well explained. They make us remember it in high school. Let’s see if we can prove it using the definitions above and the Pythagorean Theorem which states that $a^2 + b^2=c^2$

Applying the pythagorean equation $a^2 + b^2=c^2$ to the numerator, we get

And it proves that $sin^2x + cos^2x=1$.