Trigonometric Identities

1. Introduction: One Right Triangle, A Hidden Promise

You already know the six trigonometric ratios of an angle: $\sin A$, $\cos A$, $\tan A$, and their partners $\csc A$, $\sec A$, $\cot A$. You learned to read them off a right-angled triangle as "side over side".

Here is a small puzzle to start. Take any right-angled triangle. Pick one acute angle, call it $A$. Find $\sin A$ and $\cos A$ for it. Now square each one and add them.

Try this with a triangle you can imagine: sides $3$, $4$, $5$, with $A$ being the angle opposite the side of length $3$. Then $\sin A = \dfrac{3}{5}$ and $\cos A = \dfrac{4}{5}$.

So $\sin^2 A + \cos^2 A = \dfrac{9}{25} + \dfrac{16}{25} = \dfrac{25}{25} = 1$.

(One small note on notation: $\sin^2 A$ just means $(\sin A)^2$ — you square the value of $\sin A$. It does NOT mean "sin of $A^2$". Same for $\cos^2 A$, $\tan^2 A$, and the rest.)

Stop scrolling. Try it with a different triangle in your head — say sides $6$, $8$, $10$. What do you get when you add the squares?

You got $1$ again. That is not luck. It happens for every right triangle and every acute angle inside it. By the end of this lesson you will know why, and you will know two more promises just like it.