#84 5.05 Precalculus questions

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May 15th, 2015
                                cos4x - sin4x = cos(2x)
Factor the left 
side as the 
difference of 
two squares
               (cos²x - sin²x)(cos²x + cos²x) = cos(2x)

Use the identity
cos²q + sin²q = 1
to replace the second
parentheses on left
by 1.
                           (cos²x - sin²x)(1) = cos(2x)

                                cos²x - sin²x = cos(2x)
Use the identity
cos(2q) = cos²q - sin²q
to replace the left
                                      cos(2x) = cos(2x)

For all the others you first have to know that 

sin-1s means "the angle with the smallest absolute value
whose sine is s"

cos-1c means "the angle with the smallest absolute value
whose cosine is c"

tan-1t means "the angle with the smallest absolute value
whose tangent is t"

And in cases where there are two angles which are the 
smallest in absolute value, one positive and one negative,
we always choose the positive angle.


How do you find cos(sin-1(-5%2F13))

First we look only at what is in the parentheses.
We see this: sin-1(-5%2F13). This means an angle 
whose sine is -5%2F13.  The sine is negative in 
quadrants III and IV.  An angle in quadrant IV 
will have a smaller absolute value if rotated 
clockwise as a negative angle than a positive 
angle in quadrant III, so sin-1(-5%2F13) is an 
angle in quadrant IV, so we draw a radius vector 
in quadrant IV to represent the angle sin-1(-5%2F13)


Next we draw a perpendicular from the end of that
radius vector directly to the x-axis, like this:

Next we indicate the horizontal side of the 
resulting right triangle as x, the vertical side as
y, and the radius vector or hypotenuse as r.

Now that angle's sine is -5%2F13.  We know that the
sine of an angle is y%2Fr or %28opposite%29%2F%28hypotenuse%29,
so we select the numerator of the fraction -5%2F13 to be
y or the opposite, and the denominator of the fraction -5%2F13
to be r or the hypotenuse, which is the radius vector.  Since
the r or hypotenuse is ALWAYS POSITIVE, we must make the y-value
negative, which is correct because y goes downward below the
x-axis.  So we have y = -5 and r = 13.  We label these:

Now we must find x by the Pythagorean theorem

   x² + y² = r²
x² + (-5)² = (13)²  
   x² + 25 = 169
        x² = 169 - 25
        x² = 144
         x = ±Ö144
         x = ±12

and since x goes to the right we use the
positive value and x = 12. So we label
x as 12:

Now go back to the original expression:


We have drawn the angle sin-1(-5%2F13) and
found the values of x the adjacent, y the opposite,
and r the hypotenuse, and since cosine+=+x%2Fr,

cos(sin-1(-5%2F13)) = x%2Fr = 12%2F13

May 15th, 2015

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