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I just learned the very basics of Trigonometry yesterday. I can answer all the questions on the homework but feel uneasy because I still have some question about it. Here are the questions:

What does it mean to take the sin, cos, or tan of an angle? I know it has do do with the ratios but how does that apply to triangles? I just don't fully understand that concept.

Also how does the calculator know for example what the sin(53) is. When finding missing side of non special triangles, at some point I type a random number into the calculator and press either sin/tan/cos. How does it know that though?

Thanks! I'm very new to this subject and so excited to learn more.

Luke Justin
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    For your second question : https://math.stackexchange.com/questions/395600/how-does-a-calculator-calculate-the-sine-cosine-tangent-using-just-a-number. Long story short, it's a (bit complicated) numeric algorithm that involves several steps. – Matti P. Jan 12 '21 at 13:15
  • "... how does that apply to triangles?" Usually when the functions are defined as ratios they are defined by drawing a picture of a triangle and then taking ratios of the sides of the triangle. In that approach I find it hard to see how the functions could not apply to triangles. But there are other ways to do it. How was it presented to you? – David K Jan 12 '21 at 13:54
  • It was presented to me with triangles but I don't really understand what it means to take the sin/cos/tan of an angle – Luke Justin Jan 12 '21 at 14:14
  • The reason why this formula works probably won't make sense until you learn calculus, but if your want to find $\sin$ of an angle $\theta$ that is measured in degrees, then multiply that angle by $\pi/180$ (converting $\theta$ to 'radians'), and plug it into the formula $$\theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \frac{\theta^7}{7!} + \frac{\theta^9}{9!} , .$$This will give you a very good approximation. – Joe Jan 12 '21 at 18:02

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Draw a right triangle with hypothenuses of length $1$ and the given angle. The sine and cosine will be the lengths of the two other sides. This is the geometric definition. (The tangent is the ratio of these.)

There is also an analytical definition, based on the complex exponential $e^{ix}$. There are tools such as the Taylor development to evaluate the functions for a given $x$ using the four arithmetic operations. This is a broad topic. You may have a look at the very clever CORDIC method. https://en.wikipedia.org/wiki/CORDIC

  • If the OP is just learning Trig, it's unlikely the OP will be familiar with complex numbers and calculus in my opinion. – A-Level Student Jan 12 '21 at 15:18
  • @A-LevelStudent: what do you suggest ? –  Jan 12 '21 at 16:09
  • @YvesDaoust I still upvoted this post; but you could for instance mention the definition of $\sin$ when it is defined using a unit circle. I like this definition because it is powerful enough to be extended to all real numbers, while still only requiring basic geometry. Of course, once you learn that $\sin$ is an exponential function, your appreciation does become much deeper. – Joe Jan 12 '21 at 17:32
  • @Joe: that's what I did in the first paragraph ! –  Jan 12 '21 at 17:43
  • @YvesDaoust I assumed you were only talking about right angled triangles where the angle is confined between $0$ and $90$ degrees, sorry. The unit circle does generalise these ideas a fair amount. – Joe Jan 12 '21 at 17:53
  • @Joe: I would not emphasize the four quadrants for a first introduction. –  Jan 12 '21 at 17:54
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    @YvesDaoust Fair enough. – Joe Jan 12 '21 at 17:55
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    @Joe: nor would I mention complex numbers ;-) ;-) –  Jan 12 '21 at 17:56
  • I am of the firm belief that it is the best pedagogical approach to introduce $\sin$ using quaternions. – Joe Jan 12 '21 at 18:04
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    @Joe: I agree. Octonions should only be taught if the patient survives. –  Jan 13 '21 at 08:16
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Firstly, good questions! Many people (in my experience) just learning trig don't actually understand what the trig functions are, so well done for asking!

Firstly, you need to understand similarity of shapes. More specifically, when talking about right triangles, if I keep all angles in the triangle the same but I change the length of one the sides by some factor, then all the sides of the triangle will be changed by that factor.

For example, if I had a right angled triangle with side lengths $a,b$ and $c$, then if I doubled the length of the side that was $a$ to $2a$ then all the other lengths of the triangle will also be doubled. This is where trigonometric functions come in.

Consider a right angles triangle containing an angle A, as shown in the image below. enter image description here

We define the (basic) trigonometric functions as follows: $$\sin A=\frac{a}{c}=\frac{\text{Opposite}}{\text{Hypotenuse}}$$ $$\cos A=\frac{b}{c}=\frac{\text{Adjacent}}{\text{Hypotenuse}}$$ $$\tan A=\frac{a}{b}=\frac{\text{Opposite}}{\text{Adjacent}}$$

So in fact the trigonometric functions for a given angle give us the ratio of any two relevant sides of this right angled triangle. Note that because of similarity as discussed before this ratio is constant however large this right angled triangle is, as long as the angles are kept constant.

This definition cannot be extended to angles larger than $90$ degrees (or negative ones for that matter); if you want I could do this for you but it is more complicated.

As for your second question about how the calculator 'knows' what values these trigonometric functions have for a given angle, I'm afraid you'll need to wait until you learn some calculus in order to be able to understand how we can approximate values of trig functions. (We can use Maclaurin series if you know what that is; if not don't worry.)

I hope I have helped you. Please feel free to ask anything that may still be bothering you.

A-Level Student
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Draw a right triangle abc with a b the hypotenuse of length one and angle b a c your desired angle, say 53°.

Then the sine of angle bac is defined as the length of bc, the cosine as the length of ac, and the tangent as sine/cosine.

These are just the definitions of these functions of an angle.

As the computing these functions, many ways have been derived for doing it and these have been implemented and calculators and programming languages.

marty cohen
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