Friday, May 11, 2007

Trigonometry - Secant, Cosecant, Cotangent

In addition to the three basic trig functions we've already looked at (Sine, Cosine, Tangent), there are three other related functions. These are Secant, Cosecant, and Cotangent. These functions have similar meanings as the first three, in that they represent the ratios of various side lengths of a right angle triangle, and can be used to find angles or unknown side lengths. I will not go into extensive detail on these functions, as they are less commonly required, but I will show you what they mean.  Please remember to click on the Like button if this is helpful, and at the end, please hit the +1 button as well.

So far, with the help of SOHCAHTOA, we have seen that:

Sine = opposite / hypotenuse
Cosine = adjacent / hypotenuse
Tangent = opposite / adjacent

These new functions are related to the originals because they represent the inverse ratios.

Cosecant = hypotenuse / opposite... (compare to Sine)
Secant = hypotenuse / adjacent....... (compare to Cosine)
Cotangent = adjacent / opposite...... (compare to Tangent)

Also, these functions can be abbreviated: Cosecant = Csc, Secant = Sec, Cotangent = Cot.

At the middle school and high school math level, you will rarely have a need to use these functions, but it is good for you to know what they are. However, most trig problems at this stage can easily be solved with the original three functions.  Just in case, though, it's always good to know all the trig functions: sine, cosine, tangent, secant, cosecant, cotangent.

Please remember to Like and +1 this post if you found it helpful!  Thanks!

Monday, May 7, 2007

Area of a Triangle

With all this talk about triangles and their side lengths and angles, we shouldn't forget to discuss how to find the AREA of a triangle.

You are probably familiar with one formula for finding the area of a triangle:

Area = 1/2 (base)(height)

Compare this to finding the area of a rectangle:

The area of the rectangle is equal to the product of (base) x (height)..... (or length x width). However, by drawing a diagonal within the rectangle which joins two opposite corners, you can see that each newly-formed triangle is equal to half of the area of the original rectangle. Therefore, the area of a triangle is one-half the area of the rectangle, as shown by this triangle area formula. Even if you are looking at a triangle that doesn't immediately look like it is half of a rectangle, this formula still applies.

To prove it, you can draw a line in to represent the height, as I have shown here, thus creating two smaller triangles, and you can rearrange them to see that they indeed are equal to the area of half a rectangle:
That is one way to find the area of a triangle. However, if instead of base and height measurements, you are given lengths of sides or angles, this method won't work for you. In this case, you need to use a trig equation to solve for the area of a triangle.

Let's start with the first equation we had above, and modify it. By the standard trig identities, we can show that:

height = (a)(SinC)

So substituting that into our formula:

Area = 1/2(base)(height)
Area = 1/2(b)(a)(SinC)

And this is the trig formula for solving the area of a triangle!

Area = (1/2)abSinC

You can use this to find the area of a triangle where you know any two sides and the angle between them! It's that easy!

Related Posts