Calculator for finding the Area of a circle
   
 An identity is an equation which is obviously true.
Related Chapters
   
 
 Like:
X = X 3 = 3 1 hour = 60 minutes 6 = 1/2 dozen
  
 We have already seen a couple of trig identities in other chapters:
  
sin2X + cos2X = 1
 
tan X = sin X

cos X
 
sec X = 1

cos X
 
csc X = 1

sin X
 
cot X = 1 = cos X


tan X sin X
  

 In trig identity problems, you get some messy trig function equation.

 You are asked to verify that the left side of the equation

 DOES equal the right side of the equation.

 

 You might be given an equation like:

 
cosXtanXcscX = 1
  

 The strategy is to change the more complicated side of the equation

 to make it look like the simpler side.

 There is no exact strategy for solving these .

 Each problem is a puzzle with it's own route to the solution.
 
 What is generally called the "easy" way to solve these things
  is to substitute for tan, sec, csc, and cot
 so that the only terms are sine and cosine.
 
 That means using the identities we listed above.
 
 The substitutions would go like this:
 

 cosXtanXcscX = 1

  
cosX tanX cscX = 1
 
cosX sinX 1 = 1



1 cosX sinX
 
cosX sinX = 1


cosX sinX
 
1 = 1
  
 This strategy works fine, and if you like it, great.
  
 Let's try another one.
 

1

+

tan2X

= csc2X

tan2X
  
 Break it up into sines and cosines...
 

1

+ sin2X = 1

cos2X


sin2X sin2X

cos2X
  
 Generally, when we have something like 1 ± a fraction,
 you want to turn it into one term.
  
 To do this you need to have a common denominator and another way to say 1.
 We can write the
  
1 + sin2X part of the equation as cos2X + sin2X



cos2X cos2X cos2X
  
 Now we have a common denominator.
 Put this piece back in the equation.
  
cos2X + sin2X =


cos2X cos2X 1


sin2X sin2X

cos2X
  
cos2X + sin2X =

cos2X 1


sin2X sin2X

cos2X
  
 Now we get to the "Nitty Gritty."
 There are two ways you can legally "blow away" trig terms in these equations
 One is canceling (like we did in the first example).
 The other is using an identity like:
 

<some trig stuff> = 1

 
 Then we just put in the 1 for the trig stuff.
 The one totally important trig identity there is goes like this:
 

sin2X + cos2X = 1

 
 Look at what we have on the top left!
 We make that substitution. Take out the trig stuff. Put in the 1 and we get ...
  
1 =

cos2X 1


sin2X sin2X

cos2X
  
 Now we multiply the left side of the equation
 by another name for 1 to make things less messy.
 That name for 1 is cos2X/1 divided by itself.
We get:
cos2X X 1 =


1 cos2X 1



cos2X sin2X sin2X


1 cos2X
 
 Multiplying and simplifying:
cos2X 1 =

cos2X 1 1


cos2X 1 sin2X sin2X

cos2X 1
 
1 = 1


sin2X sin2X
  
 This all works, and like I said,
  if you are comfortable doing algebra on trig functions
 like we've just done you will have no problem with these things.
 
 Many people who did really well in algebra
 have trouble getting used to doing algebra on these trig function things.
 
 For these people (and I was one of them),
 there is another way to work these problems.
 
 When we started graphing triangles,
 we built a right triangle with a hypotenuse length of 1.
 

 
 The sine of angle "A" is opposite over hypotenuse, that is:
 
sin A = Y = Y

1
 
 The cosine of angle "A" is adjacent over hypotenuse, that is:
 
cos A = X = X

1
 
 The tangent of angle "A" is opposite over adjacent, that is:
 
tan A = Y

X
 
 So we have:
 
sin A (or sin X or sin (whatever)) = Y
cos A (or cos X or cos (whatever)) = X
tan A (or tan X or tan (whatever)) = Y

X
 
 The other three trig functions are just 1/(these three) so:
  
csc A = 1 = 1


sin A Y
  
sec A = 1 = 1


cos A X
  
cot A = 1 = X


Y Y

X
  
 Since cos 2X + sin 2X = 1, we can also say that X 2 + Y 2 = 1
  
 Now, we can substitute in some form of X's and Y's for all trig functions.
 We might find that makes it easier to solve these things.
 
 Let's redo the two examples that way.
 
cosX tanX cscX = 1
X Y 1 = 1


X Y
XY = 1

XY
1 = 1
   
 OK, that wasn't too bad.
 Now let's do the toughie:
 

1

+

tan2X

= csc2X

tan2X
  

1

+

Y2

=

1


X2


Y2 Y2

X2
  
 Turn the 1 at the top left into X 2/X 2 and combine terms:
  

X2 + Y2

=

1


X2


Y2 Y2

X2
  
 Use the X 2 + Y 2 = 1 thing to simplify:
  

1

=

1


X2


Y2 Y2

X2
  
 Multiply the left side by a fancy name for 1
  
X2 x

1

=

1



1

X2



X2 Y2 Y2


1

X2
  
 Combine terms and simplify:
  

X2 1

=

1


X2 1


X2Y2 Y2

X2 1
  

1

=

1



Y2 Y2
  
 We can now take this back to where the X's and Y's came from as:
  

csc2X = csc2X

  
 The steps we took with the X's and Y's
  are actually the same steps we took with the trig functions.
 
copyright 2008 Bruce Kirkpatrick