Every single time I do anything with circles/triangles I always run into the primary trig ratios. With radians, I found some hope, but it was short-lived, because yet again we needed trig functions to work with it.

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Can someone please explain how to do the primary trig functions ie. calculating sin(pi/7) without a calculator or memorization, and more importantly WHY the process is the way it is.

ik a similar question is out there, but the explanations are so damn complicated that I can only understand a few things here and there.

I"m in gr11 so please don"t use too much high-level terminology that will take me 20 google searches to understand, I"d rather save the google searches to understand the logic behind the processes.

I"m looking for something like the Taylor series


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edited Apr 22 "19 at 4:48
Allan Henriques
asked Apr 22 "19 at 3:42
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Allan HenriquesAllan Henriques
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You"re gonna have a lot of trouble figuring out weird values like $sin(29)$ or $cos(34)$. It"s only special angles that are easy to figure out. Then you have to use various formulas to fill in the gaps.

The special angles are:$$fracpi4, fracpi3, fracpi6$$

You can find $sin( heta)$ in each of the above cases using the following formulas.

$sinleft(fracpi4 ight)=frac1sqrt2$$sinleft(fracpi3 ight)=fracsqrt32$$sinleft(fracpi6 ight)=frac12$

Combine these with identities such as $sin^2( heta)+cos^2( heta)=1$, and you can get many exact values of $sin( heta)$, $cos( heta)$, and $ an( heta)$ without a calculator. There are also double- and triple-angle formulas you can use to figure out angles like $fracpi12$, and there"s the CAST rule to figure out negative angles and intermediate angles like $frac2pi3$. Look those up.

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For example, let"s say I want to figure out the value of$$x=cosleft(fracpi3 ight)$$Well, we know $sin(pi/3)=fracsqrt32$. So using $sin^2+cos^2=1$, you can get$$cosleft(fracpi3 ight) = sqrt1-sin^2left(fracpi3 ight) = sqrt1 - frac34 = sqrtfrac14 = frac12.$$

If you"re wondering about Taylor series, there"s a lot of theory behind that, but basically you need the formula$$sin(x) = sum_n=0^infty frac(-1)^n(2n+1)!x^2n+1 = x - frac13!x^3 + frac15!x^5 - frac17!x^7 + cdots$$You can use these first four terms to approximate $sin(x)$, but that can still be tough without a calculator.

P.S., re:"too much high-level terminology that will take me 20 google searches to understand" --- you have to be willing to put in the time if you want to gain a better understanding. Twenty google searches is rookie numbers. hyakkendana-hashigozake.com doesn"t happen magically over night!