Solve each of the following trigonometric equations with θ ∈ [0,2π]

Question

a) 7 tan(θ) cos(θ) = 2 cos(θ)

b) 12 csc² θ - csc θ - 1 = 0

Answer 1

7tanAcosA = 2cosA

7tanA =2

tanA = 2/7

A = tan^-1(2/7) = 15.95 degrees in quadrant I or 180+15.95 = 195.95 in quadrant III

A also = cos^-1(0) = 90 or 270

A = 15.95 or 195.95

A = about 16 or 196 degrees or 90 or 270 degrees

12csc^2(A) - cscA -1 = 0

let x = cscA

12x^2 -x -1 = 0

(3x +1)(4x -1) = 0

x =-1/3 or 1/4

cscA = -1/3 or 1/4 which is impossible, no consecant of any angle is between -1 and +1

sinA = 1/cscA = -3 or 4 which is impossible, no sine is greater than 1 or less than -1

the equation has no real solution

DNE, the solution Does Not Exist

maybe the problem was really intended to read

12cos^2A -cosA -1 = 0

then cosA = -1/3 or 1/4

A = about 180-70.5 = 109.5 degrees or 75.52 or 360-75.52 = 284.5 degrees

A= 109.5 or 250.5, or 75.52 or 284.5 degrees

A = 75.52, 109.5, 250.5 or 284.5