8.1 The Law of Sines

Hi! So this section is about the Law of Sines. Its good if you have a triangle thats oblique, or a triangle without a right angle, which is most triangles in the world.



So here we have our basic arbitrary triangle with sides a, b, and c and angles α, β, and γ. We have also drawn an altitude, h. From here, we can derive the Law of Sines.


sin α= h/b, so h= b sin α


AND


sin β= h/a, so h= a sin β


If we set the two equations equal to eachother, we get
b sin α= a sin β, or (sin α)/a= (sin β)/b


If we draw another altitude, we could also prove that (sin γ)/c is equal to these two.




So... in the end this is what we get:

If you want to use this formula, you need to know a minimum of three parts of the triangle. You can use it when you know these parts:

1. SSA - two sides and the angle opposite one of them

2. AAS/ASA - two angles and any side

To find SAS or SSS, we will have to use the Law of Cosines instead.

Theres also something important you have to keep in mind when solving with SSA. Sometimes, you solve for the sine of an angle, and you get some number between 0 and 1. There are 2 angles between 0 and 180 that have that sine, because sine functions are positive in the 1st and 2nd quadrants. This is called the ambiguous case. 

So how do you decide which angle is right? The important thing to remember is that if the measure of angle α (or β or γ) is greater than 90 degrees, than side a>b (or b>c or c>a.) If this is not the case, than angle α is acute.

That's pretty much all we covered in this chapter, so sorry the blog is so short! Bye!

7.6: The Inverse of Trigonometric Functions

Recall from Chapter 3.8 the definition of an inverse function:


Let f be a one-to-one function with domain D and range R. A function g with domain R and range D is the inverse function of f.



In Chapter 7.6, we will take the inverse of trigonometric functions sine, cosine, and tangent. These are denoted several ways:


Sine:




Cosine:
 

Tangent:






 Because you can only take the inverse of a one-to-one function, we must restrict the domain of the trigonometric functions, since they are not one-to-one.

Sine

Cosine

Tangent

Although we alter the domains, the ranges remain the same.

Sine: -1 ≤ x ≤ 1

Cosine: -1 ≤ x ≤ 1

Tangent: -∞ < x < ∞

Properties of the inverse sine:

 

       if 

       if 

Properties of the inverse cosine:

      if 

      if

Properties of the inverse tangent:

      if 

   

      if 

- Olivia Darany  

7.3

7.3  The Addition and Subtraction Formulas

Addition and subtraction formulas can help you to find exact values of trigonometric equations, solve for identities, etc.

http://math.ucsd.edu/~wgarner/math4c/textbook/chapter6/addsubformulas.htm <= good site to see how these formulas are derived

Cosine Formulas
cos(x + y) = cosx cosy - sinx siny
 cos(x - y) = cosx cosy + sinx siny


Sine Formulas
sin(x + y) = siny cosx + cosy sinx
sin(x - y) = siny cosx - cosy sinx


Tangent Formlas






Examples of how to apply the formulas:

 Cofunction Formulas:  In these formulas, 'u' is a variable that is taking place for a real number or radian measure of an angle

 -Hannah

7.4:Multiple-Angle Formulas

7.4: Multiple-Angle Formulas
Hi! In this chapter we mostly just derived a bunch of new identites (below). Hope it's helpful.

Double Angle Identities:

This identity is derived using an addition formula (sin(x+y)=sinxcosy+cosxsiny).

Like the identity above, this one also uses an addition formula.

To derive this identity, we used the formula from #2 and substituted in one minus cosine theta squared in place of sine theta squared (trigonometric identity).

We derived this identity using the addition formula for tangents.

 Power Reducing Identities:


1. To derive this identity, we used a previous identity that had sine squared in it, we then solved for sine squared.
2. This one is very similar to the first but with cosine instead.
3. For tangent squared, we substituted in equations equal to sine squared over cosine squared.

Half-Angle Identities:


1. To verify this identity we substituted u in for two theta.
2. (typo- last line should be: cos(u/2)=+/- V(1+cosu)/2 )
3. Solved by putting the identity we solved for sin(u/2) over the identity for cos(u/2).

Identities!!

7.1 Verifying Trigonometric Identities

Trigonometric Expression- Contains symbols involving trigonometric functions.


When verifying an identity, there are 3 things to keep in mind:

• Reciprocal identities




• Tangent and cotangent identities

• Pythagorean identities

It's also important to remember to cancel out factors, don't cancel out terms.

It's easier to start with the more complicated side of the equation so you can transform it into the other side using basic identities.

Here's an example that was in our homework:

(sin(θ))4 +2(sin(θ))2(cos(θ))2 + (cos(θ))4 = 1

((sin(θ))2 + (cos(θ))2) ((sin(θ))2 + (cos(θ))2) = 1 -----------> FOIL

(1)(1) = 1 --------------->pythagorean identity

This section is basically all review and we've already learned how to do this, so I'm not sure what else to say. Sorry if this is slackerish!

Megan

6.5 Trigonometric Graphs (sine and cosine)

This section was about the trigonometric graphs of sine and cosine.

To start off with, we have the graph of f(x) = sin x:

adding a value in front of sin we can stretch or compress the graph vertically, such as with

f(x) = 2sin x

we call this 'a' in the formula f(x) = a sin (bx-c)+d. b will stretch or compress vertically (although smaller numbers stretch and larger numbers compress), c will transform the graph horizontally (like b, it is counter intuitive), and d will transform vertically. For example, the graph of

f(x) = sin x+5

is shifted 5 up from f(x) = sin x, and f(x) =3 sin (1/3x-3)

is stretched vertically by a factor of 3, stretched horizontally by a factor of 3 and moved 3 to the right.

If a is negative, that will flip the graph over the midline, and if b is negative, that will flip the graph over the y-axis.

You can use the formula to draw the graph by knowing that |a|= amplitude and period = 2π/|b|.

You can also use the amplitude and period to figure out the original values of a and b in the function by taking the measurements of the amplitude and period.

All of these rules apply to cosine, but f(x) = cos x looks like

Thanks for reading, and i apologize for my abysmal post writing skills.

Dan

6.2 Identities

In this section, we learned all about identities.
Let us begin with the simplest- the reciprocal identities.

So, an example of using this would be:

There are also the quotient identities. 

And last but not least are the Pythagorean Identities!

We proved the first one in class, so here is the proof of that along with the other identities: 

And where the trigonometric functions are negative or positive: 

Alright, that's it! Sorry this took a while, I didn't get the e-mail for the blog!

MATHS IS VITAL!!!!!!!!!!

-Maggie Ridenour