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015-Circe Explains Trig, Part 1

posted Mar 21, 2012, 7:38 PM by Gregory Taylor   [ updated Mar 21, 2012, 7:39 PM ]


Primary Trigonometric Ratios 

Circe: Primary Trigonometric Ratios are really just lengths. That is, given an angle, they'll return a length.

Lyn: You mean they'll return a ratio.

Circe: Yes, but we're going to set things up so that it's a ratio of something to one. By defining the 'one', we get a length. Ergo, we start with me, the unit circle, in standard position at the origin.

Lyn: Technically, we start with a right triangle.

Circe: Look, it's a triangle with my circle's radius as a hypotenuse! Now, with an acute angle at the origin, that I shall call A.

Lyn: It's not all about you...

Circe: Well, it should be. Okay, so this is what it looks like so far... let's call the point on the circle point P. Any questions? No? Good. Now, pop quiz, how is sine defined?

Lyn: Opposite over hypotenuse.

Circe: But hypotenuse is one.

Lyn: So "sin A" is just the opposite, the vertical distance.

Circe: And therefore 'y' from the point P(x, y) on the circle. Now cosine.

Lyn: Adjacent over hypotenuse, but hypotenuse is one, so "cos A" is the horizontal distance.

Circe: And therefore 'x' from the point P(x, y) on the circle. Now tangent.

Lyn: Opposite over adjacent... which is where I get lost.

Circe: You don't see tangent on here?

Lyn: No... I mean, it's sine over cosine, so we could divide...

Circe: No! We want our ratios to be with one! You don't see tangent merely because we haven't drawn the tangent to the circle at our point P.

Lyn: Okaaaaaay....

Circe: Done. Now, a tangent will always meet a circle at what angle?

Lyn: Ninety degrees. ...oh! Of course! That means more right triangles!

Circe: (smugly) See? All about my properties. Now, using the x-axis, we have a right triangle including our same angle A as earlier. It must be a similar triangle.

Lyn: Of course, since two equal angles implies three equal angles...

Circe: Define tangent again!

Lyn: Opposite over adjacent.

Circe: But adjacent is one.

Lyn: So... so "tan A" is just the opposite, the length of the tangent from the point P down to the x-axis!

Circe: Hence the name, tangent.

Lyn: That explains so much!  Like why it has value zero at zero degrees! And why it's an impossible length at ninety degrees, since the tangent line will never reach the x-axis!

Circe: Precisely. And now we extrapolate from lengths to ratios - my radius being one doesn't matter. If you draw a bigger circle, you can merely divide every side by whatever length you've used for your radius.

Lyn: Thus the length of the tangent, from point to horizontal axis, divided by radius, is the tangent ratio! Try it at home!