0

First of all, I am not very intelligent, so the question I am about to ask would sound inane--but I am just curious, that's all.

We were always used to being attached to our calculators when it comes to utilizing the following three trig functions: Sin ($\theta$), cos ($\theta$), and tan ($\theta$). Providing an example, we were always taught that using our calculators is the solution when it comes to solving Dot Products.

Students in the Asian countries are taught without the calculator even till this day -- unless, SAT is being involved, but that is a completely different story.

I am sure there is a way to solve them by hand -- though, it may not be as precise nor easy -- and if there is a chart that can be cited, or any specific methods that can be used, please feel free to answer them below.

Thanks for your help.

Alex Jeoung
  • 23
  • I am not very intelligent is most definitely a wrong thesis, a priori. Besides that: in the past people used to have tables for $\sin$, and then, using exact values, summation and subtraction formulas (so in particular duplication), were able to find the exact values of any rational angle. Although this takes a lot of time, it works. But what I would suggest nowadays, is to work with taylor series to get fast approximations. Of course out of $\sin$ you can easily determine the values of $\cos$ as these are related by nothing but a shift of phase, and then $\tan$ is also an easy consequence. – b00n heT Jul 09 '16 at 14:09
  • I saw your profile..."I am not very intelligent". You have some inferiority complex? It really helps to talk to a psychologist. And if you are being modest then...I have nothing to say. – User Not Found Jul 09 '16 at 14:23
  • Ok so after reading the problem...I don't think that all problems can be solved by hand (my teacher is an expert at 'approximating' by hand) So, he use those approximations only when angle is close to $0$ or $\pi /2$ or something like that...if you have some special degree in mind I can try. – User Not Found Jul 09 '16 at 14:27
  • Look at the graph of trig functions with of sign changes in the quadrants. When calculator is used you can before hand expect the result approximately. – Narasimham Jun 11 '18 at 02:36

1 Answers1

0

To compute by hand, one uses a trigonometry table. The idea is relatively close to computing any logarithm from a finite table ; using the function's property.

In the case of $\sin$ and $\cos$, one has the main properties :

$$ \cos(0) = 1 \qquad \sin(0) = 0 \\ \cos\left(\frac{\pi}{2}\right) = 0 \qquad \sin\left(\frac{\pi}{2}\right) = 1 $$

$$ \cos(-\theta) = \cos \theta \qquad \sin(-\theta) = -\sin \theta $$

$$\cos(\theta) = \sin\left( \frac{\pi}{2} - \theta \right) \qquad \sin(\theta) = \cos\left( \frac{\pi}{2} - \theta \right)$$

$$\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{2}} \qquad \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \sin \theta}{2}}$$

As well as the sum and difference formulas:

$$ \sin(\alpha + \beta) = \sin \alpha \cos \beta + \sin \beta \cos \alpha \\ \sin(\alpha - \beta) = \sin \alpha \cos \beta - \sin \beta \cos \alpha \\ \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \\ \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta $$

Using a combination of these formulas allows you to compute the sine (or cosine) of a given finite angle $\theta$ within a finite time.

Calculators rather usually use power series, which are

$$ \forall x\in\mathbb{C},\, \sin x=\sum_{n=0}^{+{\infty}}(-1)^n\,{\frac{x^{2n+1}}{(2\,n+1)!}} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... \\ \forall x\in\mathbb{C},\, \cos x=\sum_{n=0}^{+{\infty}}(-1)^n\,{\frac{x^{2n}}{(2\,n)!}} = 1 - \frac{x^2}{2!} +\frac{x^4}{4!} -\frac{x^6}{6!} +... $$

Another common method is the CORDIC algorithm.

Matt
  • 608