### Trigonometric Identities - Basic Identities

Trigonometric identities are specific equalities that express one trig function in terms of other trig functions. They are fairly straightforward, but they take some work to derive them. If you are comfortable with simple derivations, you shouldn't have any problems though. Personally, I find it easier to remember the basic set of identities, and derive the more complex ones from those, rather than trying to memorize all of them... although some people are more comfortable just to memorize them.The basic identities are traditionally visualized with a triangle formed by a radius r, length x, and height y:

The basic trig definitions can easily be seen:

Sin(theta) = y/r..... opposite/hypotenuse

Cos(theta) = x/r..... adjacent/hypotenuse

Tan(theta) = y/x..... opposite/adjacent

Cos(theta) = x/r..... adjacent/hypotenuse

Tan(theta) = y/x..... opposite/adjacent

If we now apply the Theorem of Pythagoras, we can see:

r^2 = x^2 + y^2

Dividing everything by r^2 gives:

1 = (x^2)/(r^2) + (y^2)/(r^2)

1 = (x/r)^2 + (y/r)^2

Another basic relationship starts with:1 = (x/r)^2 + (y/r)^2

And then, subbing in the basic definitions, we get:

1 = [Cos(theta)]^2 + [Sin(theta)]^2

And that is the first basic identity. Nothing to it. It's proper name is the Pythagorean Trigonometric Identity. I'll rewrite it in proper notation to clean it up a bit... (Blogger is a pain with superscripts and fonts)

And that is the first basic identity. Nothing to it. It's proper name is the Pythagorean Trigonometric Identity. I'll rewrite it in proper notation to clean it up a bit... (Blogger is a pain with superscripts and fonts)

Tan(theta) = y/x

But, then sub in the Sine and Cosine definitions (isolated for x and y, respectively) to give

Tan(theta) = (r*Sin(theta)) / (r*Cos(theta))

Tan(theta) = Sin(theta) / Cos(theta)

And that's it again. This is called the Ratio Identity:

Those are now two of the simplest trig identities from which most of the others can be derived.

i like this poem.

ResponderEliminarSo you are a mathematician, too. You really intrigue me. I don't fully understand the text, but it seems to be formal. You bet it is snowing in Estoccolmo right now. I had to wear burka on my way to the store.If you live in the city of good winds, you are likely to have summer. Hasta manyana!

ResponderEliminarTrig identities were one of the most challenging task I remember in Calculus ... But somehow there seemed to be a subliminal beauty about them - I could almost feel the waves while solving the identity jigsaw puzzle. Thanks for reminding me of their beauty.

ResponderEliminar