Solve and all possible solutions 2 sin2 1 csc3 7 sin3 4 si

Solve and all possible solutions

2 sin(2) = 1

csc(3) = 7 sin(3)

4 sin() cos() + 2 sin() 2 cos() 1 = 0

csc2() = 7 cot() + 9

1) 2 sin(2) = 1

sin(2) = -1/2

2 = pi+pi/6 , 2pi -pi/6

2 = 7pi/6 , 11pi/6

General solution: = k*pi +7pi/12 , k*pi +11pi/12

2) csc(3) = 7 sin(3)

divdie both sides by cos3:

1 = 7tan(3)

tan3 = 1/7

3 = tan^-1(1/7)

3 = 0.142 , 0.142+pi

General solution: = k*pi/3 +0.142/3 = k*pi/3 + 0.047

3) 4 sin() cos() + 2 sin() 2 cos() 1 = 0

2sin(2cos +1) -1(2cos+1) =0

(2sin -1)(2cos +1) =0

sin = 1/2 ; cos = -1/2

= sin^-(1/2) = pi/6 , 5pi/6

cos = -1/2

= pi-pi/3 , pi+pi/3 = 2pi/3 , 4pi/3

general solution: = 2kpi+pi/6, 2kpi+5pi/6

= 2kpi+2pi/3 , 2kpi+4pi/3

4) csc^2() = 7 cot() + 9

use the identity: 1+cot^2 = csc^2

1+cot^2 = 7cot +9

cot^2 -7cot - 8 =0

cot^2 -8cot +cot - 8 =0

cot( cot -8) +1(cot -8) =0

(cot +1)(cot -8) =0

cot = -1 ----> = cot^-1(-1)

= pi-pi/4 , 2pi-pi/4 = 3pi/4 , 7pi/4

cot = 8 ---> = 0.124 , pi +0.124 = 0.124 ,3.27 rad

General sol: = kpi +3pi/4 , kpi+7pi/4 , kpi +0.124 , kpi +3.27