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1. ## Law of Sines?

A friend of mine made this picture to illustrate the law of sines.

I kinda see a pattern, but not completely...and I don't know how this knowledge would be useful.

2. What are you crazy???? The Law of Sines and Cosines are EXTREMELY useful. It's not even constrained to finding the lengths of things. Once you get into vectors, you can use these to find velocities, accelerations, many things. Things don't even have to be triangular from the start either. Given like a rectangle, you can draw lines from opposite corners to make triangles. Trust me, it's something you'll wanna know.

3. Having never gotten beyond basic Algebra in high school, and being self taught on computers, I learned to use sin & cosine in a way that is backwards from the method that is usually taught. I learned about their practical application before I learned about their relationship to each other, and to the circle.

I first encountered sine waves when using analog music synthesizers in the early 80s. It was one of the settings that the oscillators had, along with saw tooth waves, triangle waves and square waves. I knew what a sine wave looked like, and what it sounded like, but that was about it.

When I started using computers, I experimented with drawing sine waves in BASIC programs. At first, that's all I thought they were good for, for making that 'sine wave' shape.

10 FOR X = 1 to 100
20 PLOT X, SIN(X*.1)
30 NEXT X

Pretty soon I figured out that the SIN wave repeated itself every 2 PI, so if I set my scaling factor just right, I could get a perfect sine wave:

10 FOR X = 0 to 100
20 PLOT X, SIN(X*2*PI/100)
30 NEXT X

And I could get two sine waves if I doubled the scaling factor (or frequency).

10 FOR X = 0 to 100
20 PLOT X, SIN(2 * X*2*PI/100)
30 NEXT X

Then I started screwing around with COS, and realized it was an out of phase sine wave.

10 FOR X = 0 to 100
20 PLOT X, COS(2 * X*2*PI/100)
30 NEXT X

I figured out that I could get the same result using SIN(X+PI/2).

COS(X) = SIN(X+PI/2)

When I tried plotting a SIN in X and a COS in Y, I saw a perfect circle staring back at me.

10 FOR X = 0 to 100
20 PLOT SIN(X*2*PI/100), COS( X*2*PI/100)
30 NEXT X

From this I figured out the basic forumla for tracing the outline of a circle.

X = COS(A)*R
Y = SIN(A)*R

It was a short jump from this to drawing various kinds of more interesting figures, by playing with the constants in the above formula. Soon I was making hexagons, octagons, stars, sunbursts, spirograph-like designs and yin-yang symbols.

Eventually I found uses for the related trigonometry functions, TAN, ACOS, ASIN, ATAN, not to mention LOG, EXP, POW and so on.

My point is that the best way (for me) to learn about these functions was to PLAY with them. There is no substitute for hands-on experience.

See my website (link below) for lots of examples of sines and cosines in action (in actionscript, no less). Especially in the MANDALA chapter.

- Jim

4. Basically:

(SinA)/a = (SinB)/b = (SinC)/c

Where on a right triangle, angle A is opposite side a, B is opposite b, and C is opposite c.

5. Originally posted by prthealien
Basically:

(SinA)/a = (SinB)/b = (SinC)/c

Where on a right triangle, angle A is opposite side a, B is opposite b, and C is opposite c.
No no no. That's what beautiful about the law of sines and cosines. They can be used for ALL triangles, not just right.

6. Suppose we have a triangle with two sides A and B. The oppositve angles are b and a.
What is the "height" of this triangle ?
height=A*sin(b); Also height=B*sin(a);
Since the hieght is the same, the equation is A*sin(b)=B*sin(a);
This is how we get the forumla: sin(a)/A=sin(b)/B;
If we take another angle c and the opposite side C, the forumla turns to be sin(a)/A=sin(b)/B=sin(c)/C;

7. Sine wave enthusiasts will enjoy this Interactive Spyrograph, a recent addition to the Bestiary.

- Jim

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