someone please help me solve these three trig equation..I cant figure it out?

Question

I was doing my Hw and I cant seem to solve these three trig equation...please help

1.) 2sinx +1 =csc x ; interval 0< or = to x
2.)Cos X= Sin X ; interval 0< or = to x < pi

3.)12Cos2x-16sin^(2)X=0 ; interval 0< or = to x < pi

Answer 1

1]

2 sin x + 1 = cosec x
2 sin x + 1 = 1 / sin x
2 sin^2 x + sin x = 1
2 sin^2 x + sin x - 1 = 0
Let (sin x) be y,
2 y^2 + y - 1 = 0
(2y - 1) (y + 1) = 0
2y - 1 = 0 or y + 1 = 0
y = 0.5 or y = -1
sin x = 0.5 or sin x = -1(rejected)
x = sin^-1 (0.5)
basic angle x = 30 deg
Since sin x is positive, x is in 1st and 2nd degree:
x = 30; 180-30
x = 30; 150


2]

cos x = sin x
1 = sin x / cos x
1 = tan x
x = tan^-1 (1)
basic angle x = 45 deg
Since tan x is positive, x is in 1st and 3rd deg:
x = 45; 180+45
x = 45; 225

Answer 2

i visit anticipate that sin^2 x is (sin(x))^2, although right here approach could artwork even no depend if it extremely is sin(2x), in uncomplicated terms use a trig id to shrink sin(2x) to 2sin(x)cos(x), to remedy the place the two graphs intersect first substitute y = sin(x) into the 2nd equation, giving sin(x) = 2 (sin(x))^2 subtract sin(x) from the two facets 0 = 2 (sin(x))^2 - sin(x) ingredient out a sin(x) 0 = sin(x) (2sin(x) - a million) while the two sin(x) or 2sin(x) - a million is comparable to 0 the two graphs will intersect. sin(x) = 0 while x = PI * ok the place ok is any integer sin(x) = a million/2 while x = PI/6 + 2 * PI * ok the place ok is any integer Now only plug x decrease back into the unique equation to discover the corresponding y fee. Notes: x is given in radians, the corresponding diploma values are only x*a hundred and eighty/PI, and PI is the mathematical consistent pi approximately 3.14

Answer 3

2 sin x+1 = 1/sin x so 2sin^2x+sin x -1 =0 call sin x = z and you get a 2nd degree equation in z (Than use formula)
2 tan x= 1 x= pi/4
cos 2x = cos^2-sin^2 =1-2sin^2x
so 12-40sin^2x= 0
sin^2 x= 10/3 impossible as sin is between -1 and 1