Calculator for finding the Area of a circle
   
 To solve for the unknown angles or side lengths
Related Chapters
   
 in these triangles:
We need another shortcut.
 
 The new short cut looks kind of like the Pythagorean theorem.
 Actually, it IS the Pythagorean theorem,
 modified to work with any triangle not just right triangles. 
 
 Here's how it goes:
 For any triangle
 The lengths of the three sides, a, b, and c, and one angle, C,
 show up in this formula.
 You need to know three of the four things to find the fourth.
 You can set up the labels any way you want on a triangle
 as long as an angle and the side opposite that angle use the same letter.
 That means we can write this formula as any of these:
 
The point is, 
 there is nothing special about any of the sides or angles in the triangle.
 You can start with whatever sides and angle you like depending
 on what missing info you are looking for at the moment.
 
 Example:
 Label it:
 Now solve for something, say angle C:
 
  
 Subtract 74 from each side:
  
   
 Divide each side by -70 and simplify:
  

 Now find the angle that has a Cosine of 0.92857:

 

 Remember, Cos-1 means: "What angle has a cosine of"

 

 Filling in the new thing we know we have:

 OK. Now we have enough info to use the trick with the sines from the last page, 
 (the Law of Sines), if we want.
 Most people think that one is easier to use.
 For fun, let's use the new trick again.
Did I say FUN! Yikes, I should get a life, eh?
 Here we go. This time, let's find the measure of angle B.
 To do this, use the version of the equation with the Cos B in it.

 Subtract 58 from each side:

   

 Divide each side by -42 and simplify:

  
 Now find the angle that has a Cosine of 0.78571:
 Filling in the new thing we know we have:
 Now we have one more thing to find.
 We could use the Law of Sines from the last page
 or the new equation from this page, 
 but we could also use the "Angles inside the triangle add up to 180° " thing
 and it is WAY easier, so:
 
 Fill in that and we're done:
 
 Example:
 
 Here's one where we know the lengths of two sides
 and the measure of the angle between them:
 
 We can't use the Law of Sines from the last chapter on this one.
 Can you see why?
The reason is that any pair of angle and side across has an unknown.
 That means when we write the equation it will have TWO unknowns.
 That's one more than we can deal with in an equation like that.
 Here we go. Label the sides and angles:
 Choose a form of the equation.
 We have sides b and c and angle A so we need the version
 that includes those three:

 Look up and insert the Cosine of 35° in the equation(0.8192) and carry on:

 Filling in the new thing we know we have:
 Usually in math when we find square roots, 
 we need to think about both the positive and negative roots. 
 The thing with triangles is that negative lengths don't really mean anything 
 so we can ignore the negative root here.
 
 Now we have the measure of an angle and the length of the side across from it. That means we COULD use the Law of Sines to find the rest of the missing info.
 
 
 But since the Law of Sines was LAST page, 
 we will carry on with our new formula.
 
 Let's find the measure of angle C next.
 Since we want to find angle C,
 we need the version of the equation that has Cos C in it:

 Filling in that, we have:

 Just one more thing to find, the measure of angle B.
 Let's get lazy and use the "Angles add up to 180°" thing.
 

 Fill that in and we're done:

 The official name of the equation we have been using in this page

 SHOULD have been something to do with the Pythagorean Theorem. 

 It isn't. 

 Math types decided to call it THE LAW OF COSINES. 
 I guess since there was a cosine in the equation that made sense.
 

 It's quite a stretch if you ask me.

 

 They didn't.

 

 There is one case where three pieces of information is just not enough.

 That happens when we have the length of two sides
 and the measure of an angle that is NOT between them.
 Like this:
  
 The reason we can't solve this one is that we can make 
 two completely different looking triangles
 that have the angle and sides like we have here.
 We can make the one we have above,
 and we can also make this one:
 
 The way to look at this is that there are two different places
 that the side with length of 6 can attach to make this triangle work out
 with the numbers we have:
 Since we can't tell which one of these it is from the information
 (and you can't trust a picture on a math test),
 we can't solve this one.
 It is called "The Ambiguous Case"

 Now I have some more bad news. 

 We haven't gone through all the trig stuff we need to

 to show you how we know the Law of Cosines really works. 
 In other words, we can't do a PROOF yet even if we wanted to.
 
 Yeah, I know you're really broken up about that one.
 
 We will get to it, eventually.
 
 The next important thing we need to talk about 
 is trig without triangles, just angles.
 
 The place to begin that, is to talk about graphing this stuff.
 
copyright 2008 Bruce Kirkpatrick