Use a halfangle formula to find the exact value of the expre

Use a half-angle formula to find the exact value of the expression. sin 5 pi/12 sin 105 degree Find all solutions of the equation. sin 4x = squareroot 3/2 cos 2x = squareroot 2/2 Solve the equation on the interval (0, 2 pi). cos^2 x + 2 cos x + 1 = 0 2 sin^2x = sin x 2 cos^2 x + sin x - 2 = 0

Solution

31) We have sin(1/2(x)) = +- sqrt(1-cosx)/2

Now consider x = 5pi/6

Then sin(5pi/12) = + - sqrt(1-cos(5pi/6))/2

=> sin(5pi/12) = + sqrt(1-(-sqrt3/2)/2)

=> sin(5pi/12) = +sqrt((2+sqrt3)/4) = +(1/2)sqrt(2+sqrt3)

32) sin(105⁰) = sin(180-75) = sin(75⁰) = sin(5pi/12) = (1/2)sqrt(2+sqrt3)

33) sin4x = sqrt(3)/2

=> sin4x = sin(pi/3), sin(pi-pi/3), sin(2pi+pi/3), sin(3pi+pi/3),..

=> 4x=pi/3, 2pi/3, 7pi/3, 10pi/3,..
=> x=pi/12, pi/6, 7pi/12, 5pi/6,..

34) cos2x = sqrt2/2

=> cos2x = cos(pi/4), cos(2pi-pi/4),cos(2pi+pi/4),cos(4pi-pi/4),

=> 2x=pi/4, 7pi/4, 9pi/4, 15pi/4

=> x=pi/8, 7pi/8, 9pi/8, 15pi/8

35) cos²x + 2cosx + 1 = 0

Let y = cosx

Then y²+2y+1=0

=> (y+1)²=0

=> y=-1

=> cosx = -1

=> cosx=cospi

=> x=pi

36) 2sin²x = sinx

=> 2sin²x-sinx = 0

=> sinx(2sinx-1)=0

=> sinx = 0 and sinx=1/2

=> x=0,,pi and x = pi/6, pi-pi/6 = 5pi/6

Hence x=0, pi, pi/6, 5pi/6

37) 2cos²x +sinx-2=0

2(1-sin²x)+sinx-2=0

=> 2-2sin²x+sinx-2=0

=> -2sin²x+sinx=0

=> sinx(-2sinx+1)=0

=> sinx=0 or sinx=1/2

=> x=0, pi, pi/6, 5pi/6