Math Trig Problem. Help Please?

Question

Could someone help me solve this problem. I'm having trouble with it.

Solve for all real values of x.

2 sinx + cscx = 3

Thanks. Solve in steps please

Answer 1

2 sinx + cscx = 3

=>2sinx + (1/sinx) = 3

=>2sin^2(x) + 1 - 3 sinx = 0

=>2sin^2(x) - 3sinx + 1 = 0

=>2sin^2(x) - 2sinx - sinx + 1 = 0

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

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

sinx = 1 or 1/2

when sinx = 1

x = pi/2, (4n+1)pi/2 where n is any positive whole number

when sin x = 1/2

x = pi/6, 5pi/6, (pi/6)(12n + 1)) where n = all positive whole

numbers

all the values of x are

pi/2, pi/6, 5pi/6, (4n+1)pi/2 and (12n+1)pi/6

where n is any positive whole number

Answer 2

cscx = 1/sinx so after substituting you get

2 sin x + 1/sin x = 3
Now multiply through by sin x to clear the denominator

2 sin ^2 x + 1 = 3 sin x
Get = 0
2sin^2 x - 3 sin x + 1 = 0

Factor
(2 sinx -1)(sin x -1) = 0

solve each factor = 0

sin x = 1/2 or sin x = 1
x = 30 degrees , 150 deg + 360k or x = 90 degrees + 360 k

or in radians: pi/6 + 2k pi, 5 pi/6 + 2 k pi, or pi/2 + 2kpi

Answer 3

csc x = 1/sin x

2 sin x + 1/sin x = 3

multiply all by sin x to clear fraction

2 sin^2 x + 1 = 3 sin x

make it = 0

2 sin^2 x - 3 sin x + 1 = 0

Factor: (2 sin x - 1)(sin x - 1) = 0

so sin x = 1/2 or sin x = 1

sin x = 1/2 if x = 30 or 150 or 360n + 30, 360n + 150
sin x = 1 id x = 90 or 360n + 90

where n is any integer. Or convert to radians if needed.

Answer 4

2sinx + 1/sinx = 3
2sin^2x -3sinx + 1 = 0
(2sinx -1)(sinx -1) = 0
sinx = 1/2 or sinx = 1

x = pi/6, 5pi/6 or pi/2

Answer 5

attempt this sin120/cos45= tan(4x+ninety). then take the inverse tangent. ought to grant you with tan^-a million(sin120/cos45) = 4x+ninety. you should have an total form for the left aspect then do exactly elementary algebra and subtract ninety from the left aspect and then divide through 4 and also you should have your answer.