Circle all expressions that are equivalent to sin 3a 3 sin a

Circle all expressions that are equivalent to sin 3a. 3 sin a cos a 3 sin a cos^2a - sin^3 a 2 sin a cos^2 a + sin a - 2sin^3 a 3 sin^2 a cos^3 a 4 sin a cos^2 a - sin a PlusminusSquareroot1-cos6a/2 17. Circle the expression that is equivalent to tan 5pi/12. 2 + Squareroot3 Squareroot3-2 2-Squareroot3 none of these Given that the exact value of cos pi/5 = 1 + Squareroot5/4,for each question calculate the exact value of the given expression. You may leave nested radicals in your answer. 18. cos 2pi/5 19. cos pi/10

Solution

16) sin3a

sin3a = sin(2a+a) = sin2a*cosa +cos2a*sina

=2sina*cosa*cosa +cos2a*sina

= 2sina*cos^2a + sina*cos2a

= 2sina*cos^2a + sina( 1-2sin^2a)

=2sina*cos^2a +sina -2sin^3a

Further, use cos2x = 1-2sin^2x

Then, cos6a = 1 -2sin^2(3a)]

sin3a = +/- sqrt{[ 1- cos6a]/2}

17) tan(5pi/12) = tan(10pi/12)

= tan(pi/4 +pi/6)

tan(x+y) = (tanx +tany)/( 1-tanxtany)

tan(pi/4 +pi/6) =[ tanpi/4 +tanpi/6]/[ 1- tanpi/4*tanpi/6]

= [ 1 +1/sqrt3]/[ 1- 1/sqrt3]

= [sqrt3 +1]/[sqrt3 -1]

ratinomalise denominator:

=  [sqrt3 +1]^2/(3 -1 )

= ( 3 +1 +2sqrt3)/2

= (4 +2sqrt3)/2

= 2 +sqrt3