exact solutions

Question

Find all exact solutions on the interval 0 ≤ x < 2π. (Enter your answers as a comma-separated list.)

cot(x) + 5 = 4

csc²(x) − 8 = −4



Answer 1

cotx+5=4

cotx=-1

tan=sin/cos

cot=cos/sin

at what angles are sin and cos opposites ?

look on the unit circle

at 135° cos=-√2/2 and sin=√2/2

at 315° cos=√2/2 and sin=-√2/2

x=135° and x=315°

csc^2x-8=-4

cscx^2x-8+4=0

csc^2x-4=0

factor

(cscx-2))cscx+2)=0

cscx=2

cscx=-2

cscx=1/sinx

therefore...

1/sinx=2 and 1/sinx=-2

so,....

sinx=1/2 and sinx=-1/2

look on the unit circle

sin is 1/2 at 30° and at 150°

sin is -1/2 at 210° and at 330°

x=30°, x=150°, x=210°, x=330°