Notes 1/26 - Law of cosines

1/26 - And we're back to school. . . Today we took notes on "the law of cosines." This formula allows you to solve for any triangle (find all values for angles and sides) if at least 3 of these values are known. The law of cosines works with any type of triangle - not just right triangles. One thing to remember in notation is that uppercase letters refer to angles and lowercase to sides. In solving for a triangle using the law of cosines it does not matter which value you look for first, as long as you can find the others taking that approach. Also, remember to check if the triangle is even possible in the first place. The following is the formula for law of cosines:

a^2 = b^2 + c^2 - 2bc*cosA

Today we studies the Double Angle ID's: sin(2A)=2sinAcosA and cos(2A)=cos^2A-sin^2A. Once you solve for both sin(2A) and cos(2A), you can solve for tan(2A)=sin(2A)/cos(2A). It's important to know that for that the 2 in 2 sinAcosA isn't the same as the 2 in its equivalent sin(2A). Its derived from the sum of the 2A represented in the equation 2(sinAcosA).

12/2/09 Sum/Difference Identities

In class on December 2nd we learned about the Sum and Difference identities. (Note: a large portion of these notes show how we derived the identities given angles and prior knowledge of the unit circle.)

The identities allow us to find the sine, cosine, and tangent of angles we previously could not (ie: 15 degrees), by adding or subtracting (ie: 60-45=15) using angles for which we can already find the trig functions.

Coefficients of Angles (Trigonometry)

We discussed in class that normally when using trig identities to the meaasure of an angle, you come out with two possible angles. However, a coefficient of an angle produces that double that number of solutions.
girl in the hall: "It took me three minutes to get ready today"
MV: "I can tell"

Notes: 11/30

Sorry it took me this long to post the notes. I had a few problems. Anyway, today in class we learned how to solve trig equations by using 3 main steps:
1.Get the 1st answer from the inverse trig function
2.Then the second answer from the reflection of the trig ID (Ex: Tan= 180 or 2pie+the angle)
3.Add or subtract 2 pie or 360 (depending on whether you are in degrees or radians) to both of your answers from 1 and 2 to get as many answers as needed.

11/23 Notes

In class, we worked with trigonometry identities. When you have a trig equation, you want to first look to do algebra: factor, expand, or get a common denominator. Second, look for trig IDs to use. We learned definitions ie sec(x)=1/(cos(x)). We also looked at the pythagorean IDs: cos^2(x) + sin^2(x)=1, 1+tan^2(x)=sec^2(x), cot^2(x)+1=csc^2(x). We also studied complement IDs (ex. cos(x)=sin((pi/2)-x))and reflection IDs (ex. sin(x)=(pi-x)). We applied these IDs to simplify equations and expressions.

11/17 class notes

Today in class we talked about inverse trigonometric functions, like arcsin etc. The main concept we covered was the ambiguity of the number of solutions for these functions. In other words, how do we restrict arcsin etc. to only produce one solution? (for example if you plugged in arcsin= .5 to your calculator how could you insure that it would give you the answer 30 degrees rather than 390, - 330, etc?) Well what we do is restrict the domain and range. In the chart in the notes it lists the different domain and range restrictions for the different functions. The reason we have to insure they only have one solution is that if they had multiple they would not be functions anymore (aka more than one output).