# Thread: Trig identities with special triangles

1. ## Trig identities with special triangles

the answer in the back of the book is 2-root(3) the answer i keep getting is 2+root(3)

Find the exact value of each by choosing the values for angle a and b and applying an appropriate sum or difference identity:

tan 15

tan 15 = tan (pi/4 + pi/6)

tan (A+B)= sin( A+B) / cos (A+B)

so:

sin (A+B) = sin pi/4 cos pi/6 + cos pi/4 sin pi/6
(1/root(2))(root(3)/2) + (1/root(2))(1/2)
= (root(3) +1)/(2root(2))

cos (A+B) = cos pi/4 cos pi/6 - sin pi/4 sin pi/6
= (1/root(2))(root(3)/2) - (1/root(2))(1/2))
= (root(3)-1)/(2root(2))

tan (a+b) = ((root(3)+1)/(2root(2))/((root(3)-1)/(2root(2))
= (root(3)+1)/(2root(2) times (2root(2))/(root(3)-1)
(cancel out the 2root(2))
=(root(3)+1)/(root(3)-1) <---- multiply both sides by conjugate (root(3)+1)
=(3+2root(3) +1)/ 3-1
=(4+2root(3))/2
simplify
=2+root(3)

2. Originally Posted by tmas
the answer in the back of the book is 2-root(3) the answer i keep getting is 2+root(3)

Find the exact value of each by choosing the values for angle a and b and applying an appropriate sum or difference identity:

tan 15

tan 15 = tan (pi/4 + pi/6)

tan (A+B)= sin( A+B) / cos (A+B)

so:

sin (A+B) = sin pi/4 cos pi/6 + cos pi/4 sin pi/6
(1/root(2))(root(3)/2) + (1/root(2))(1/2)
= (root(3) +1)/(2root(2))

cos (A+B) = cos pi/4 cos pi/6 - sin pi/4 sin pi/6
= (1/root(2))(root(3)/2) - (1/root(2))(1/2))
= (root(3)-1)/(2root(2))

tan (a+b) = ((root(3)+1)/(2root(2))/((root(3)-1)/(2root(2))
= (root(3)+1)/(2root(2) times (2root(2))/(root(3)-1)
(cancel out the 2root(2))
=(root(3)+1)/(root(3)-1) <---- multiply both sides by conjugate (root(3)+1)
=(3+2root(3) +1)/ 3-1
=(4+2root(3))/2
simplify
=2+root(3)
15 = 45 - 30 NOT 45 + 30 (and certainly not for that matter pi/4 + pi/6).

3. Another alternative is the half angle identity

$\displaystyle \tan{\frac{\theta}{2}} = \frac{1 - \cos{\theta}}{\sin{\theta}}$.

Here you would make $\displaystyle \theta = 30^{\circ}$ or $\displaystyle \theta = \frac{\pi}{6}^C$ (like Mr F said, don't switch between degrees and radians - make a choice and stick with it).