Mysterious Art of Handling Trigonometric Ratios

Weeks back, when I first had my trigonometric lesson, I was taught the simple basic identities of trigonometric ratios for special angles such as $30^0$ , $45^0$ and $60^0$ not forgetting right angles and zeroes too. Then it hit me, while I may had basic prior knowledge of trigonometry back then before the new lesson, I wondered how did trigonometric ratios such as $tan, cos and sin$ even came about in the first place? Is there a certain pattern or formula or any other that may allow me to retrieve the values of Trigo. ratios for any angles quickly without the use of calculator? Is it possible to mentally calculate trigonometric ratios for any angle as fast as multiplying numerical values such as $5\times 7$ ? Though I know it is pretty hard to answer it now or there may not even be an answer to these questions. However, I believe in time to come, the secret to mental calculation of trigo. ratios may be discovered. I hope the note isn't overly-ambitious on the impossible....

#Geometry

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`2 \times 3`	$2 \times 3$
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Comments

Use the infinite expansion of both $\sin$ and $\cos$ if you don't have calculator. At least you would get approximate answer using only a few terms. For problems, usually the ratios can be found using sum and difference, double angle and half angle, etc.

Marc Vince Casimiro - 6 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$