Thread: I need help

Use identities to find each exact value. (Do not use a calculator).


cos(-pi/12)

Hello,

Originally Posted by norwoodjay

Use identities to find each exact value. (Do not use a calculator).


cos(-pi/12)

We know that cos(-x)=cos(x)

Thus cos(-pi/12)=cos(pi/12).
pi/12=(pi/6)/2.
--> cos(-pi/12)=cos((1/2) pi/6)

We know that cos(2x)=2cos²(x)-1. Then cos(x)=2cos²(x/2)-1 --> cos(x/2)=+ or - sqrt[(cos(x)+1)/2]

therefore cos(-pi/12)=+ or - sqrt[(cos(pi/6)+1)/2]
Since -pi/12 is in the fourth quadrant, the cosine is positive.

--------> cos(-pi/12)=sqrt[(cos(pi/6)+1)/2]

(and you should know cos(pi/6))

cos(-pi/12)=sqrt(sqrt(3)+2)/2

cos(-pi/12) = cos(pi/12)

Why? Because cosine is an even function. Now, let's rewrite pi/12 as:
pi/4 - pi/6

So we got now cos(pi/12) = cos(pi/4 - pi/6). Now, recall the sum and difference identites for cosine:

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

Let's use it then!

cos(pi/4 - pi/6) = cos(pi/4)cos(pi/6) + sin(pi/4)sin(pi/6)

Now, since they asked you not to use the calculator, you must use the special triangles
1. pi/4 - pi/4 - pi/2 (45-45-90)
2. pi/3 - pi/2 - pi/6 (60-90-30)

cos(pi/4 - pi/6) = ( (sqrt(2)/2) x (sqrt(3)/2) ) + ( (sqrt(2)/2) x 1/2 )
= (sqrt(6)/4) + (sqrt(2)/4)

Final answer: ( sqrt(6) + sqrt(2) )/4

Note that our solutions are equal !

( sqrt(6) + sqrt(2) )/4

sqrt(sqrt(3)+2)/2


See in this post of earboth the identity : http://www.mathhelpforum.com/math-help/32751-post3.html (the identity can easily be shown by squaring both sides)

Originally Posted by norwoodjay

Use identities to find each exact value. (Do not use a calculator).


cos(-pi/12)