2sin(x)-(2+sqrt(3))=-sqrt(3)csc(x)

Solution

2 sin (x) - (2 + 3) = - 3 csc (x)

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{π}{2} + 2 πn

Solution steps

2 sin (x) - (2 + 3) = - 3 csc (x)

Subtract

- 3 csc (x)

from both sides

2 sin (x) - 2 - 3 + 3 csc (x) = 0

Rewrite using trig identities

- 2 + \frac{2}{csc ( x )} - 3 + csc (x) 3 = 0

Solve by substitution

csc (x) = \frac{2 3}{3}, csc (x) = 1

csc (x) = \frac{2 3}{3} : x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

csc (x) = 1 : x = \frac{π}{2} + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{π}{2} + 2 πn

7 cos^{2} (θ) + 6 sin (θ) - 10 = - 4

3 (2 sin (x) - cos (x)) = 2 (sin (x) - 3 cos (x))

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

tan (θ) = \frac{7}{6}

2 sin (x) + 5 = 6