How to you use the trigonometric functions without a calculator?

Question

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.

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

Answer 1

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: $$\frac{\pi}4, \frac{\pi}3, \frac{\pi}6$$

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

• $\sin\left(\frac{\pi}4\right)=\frac{1}{\sqrt{2}}$
• $\sin\left(\frac{\pi}3\right)=\frac{\sqrt{3}}{2}$
• $\sin\left(\frac{\pi}6\right)=\frac{1}{2}$

Combine these with identities such as $\sin^2(\theta)+\cos^2(\theta)=1$, and you can get many exact values of $\sin(\theta)$, $\cos(\theta)$, and $\tan(\theta)$ without a calculator. There are also double- and triple-angle formulas you can use to figure out angles like $\frac{\pi}{12}$, and there's the CAST rule to figure out negative angles and intermediate angles like $\frac{2\pi}3$. Look those up.

For example, let's say I want to figure out the value of $$x=\cos\left(\frac{\pi}3\right)$$ Well, we know $\sin(\pi/3)=\frac{\sqrt{3}}2$. So using $\sin^2+\cos^2=1$, you can get $$\cos\left(\frac{\pi}3\right) = \sqrt{1-\sin^2\left(\frac{\pi}3\right)} = \sqrt{1 - \frac{3}4} = \sqrt{\frac{1}4} = \frac{1}2.$$

---

If you're wondering about Taylor series, there's a lot of theory behind that, but basically you need the formula $$\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1} = x - \frac{1}{3!}x^3 + \frac{1}{5!}x^5 - \frac{1}{7!}x^7 + \cdots$$ 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. Math doesn't happen magically over night!

Answer 2

Finding sine or cosine for any angle $\alpha$ can be always reduced to finding a sine or cosine for some angle $\beta \in [0, \pi/2)$.

Finding sine or cosine for such (acute) angle $\beta$ can be always replaced with finding a sine or cosine for some angle $\gamma\in[0, \pi/4)$. For example: $\sin\frac{3\pi}{8}=\sin(\frac\pi2-\frac\pi8)=\cos\frac\pi8$

All you need is a good formula to calculate $\sin x$ and $\cos x$ for $x\in[0, \pi/4)$. For most practical purposes just a few items of the Taylor series will suffice:

$$\sin x\approx x-\frac{x^3}6\tag{1}$$

$$\cos x\approx 1-\frac{x^2}2+\frac{x^4}{24}\tag{2}$$

For the fiven range the maximum error of (1) is 0.00245413 or 0.2%. The maximum error of (2) is 0.000322426, almost neglectable for most practical purposes.

Answer 3

The Taylor series is can be used to find very accurate estimates of trig functions.
However if you want to find $\sin \frac{\pi}{7}$ then it isn't going to get you very far.

$\sin x = x - \frac {x^3}{3!} + \frac {x^5}{5!} - \frac {x^7}{7!} + \cdots$

If $x$ is an irrational (or trancendental) number like $\frac {\pi}{7}$ you are not going to get very far trying to work this out by hand.

However, I suspect you haven't taken calculus yet, so, teaching you the Taylor series is pointless, as you don't know the math behind it.

The precise values of these trigonometric functions is not so important. It is far less important than it is to understand what these functions represent. And, measuring the lengths of sides of triangles is ultimately one of the less interesting properties, even if this is how they are traditionally introduced.

Answer 4

We say that going all the way around the unit circle is to go "2*pi." That is, going "pi" around the circle is to go half way around and going pi/7 is to go 1/14 of the way around the whole circle.

If we say that our radius is 1, we can solve for our respective x and y values along the circle. Our x value is the "cosine," while our y value is the "sine." Both are made-up, mis-spelled and mistranslated arabic words that have no other meaning; sin and cosine are just 2 of the six ratios we can make by comparing the 3 sides of a right triangle.