4sin2(x) + 15sin(x) - 4 = 0

Question

Use inverse trigonometric functions to find the solutions of the equation that are in the interval [0, 2π). You may enter an exact answer or round your solutions to four decimal places. (Enter your answers in a comma-separated list.)

 

4sin2(x) + 15sin(x) - 4 = 0

Answer 1

FACTOR!  (4sinx - 1)(sinx + 4) = 0

 

The second of these factors cannot be zero; the first factor is zero if sin x = ¼.

 

use your calculator to find what radian value(s) of x have sine equal to one fourth.

Answer 2

The expression can be factored giving:

 

(sin(x) +4)(4 sin(x) - 1)  = 0

 

Since sin(x) cannot equal 4,  this implies

  4 sin(x) -1 = 0.    Thus  sin(x) =  1/4

 

 x = arcsin(1/4) (1st quadrant)

 x =  pi - arcsin(1/4)    (2nd quadrant)

 

Numerically,   x =   0.2527 ,   2.8889