How do you find all solutions to sin (2x + 1) = 0.2?

2 Answers

It has infinite set of solutions

Explanation:

The general solution is given as follows
#\sin(2x+1)=0.2#
#2x+1=2n\pi+\sin^{-1}(0.2)# OR #2x+1=2n\pi+\pi-\sin^{-1}(0.2)#
#x=\frac{2n\pi-1}{2}+\frac{1}{2}\sin^{-1}(0.2)# OR #x=\frac{(2n+1)\pi-1}{2}-\frac{1}{2}\sin^{-1}(0.2)#
where, #n# is any integer i.e. #n=0, \pm1,\pm2, \pm3, \ldots#

Jun 19, 2018

#x = 55^@57 + k360^@#
#x = 337^@11 + k360^@#

Explanation:

1 is expressed in radians. We can convert it to degrees for easier solving.
#pi = 3.14# --> #180^@#
1 radian --> #180/3.14 = 57^@32#
sin (2x + 57.32) = 0.2
Calculator and unit circle give 2 solutions for (2x + 57.32):
a. #(2x + 57.32) = 11^@54#
#2x = 11.54 - 57.32 = - 45^@78#
#x = - 22^@89#, or #x = 360 - 22.89 = 337^@11# (co-terminal).
b. #(2x + 57.32) = 180 - 11.54 = 168.46#
#2x = 168.46 - 57.32 = 111.14#
#x = 55^@57#
For general answers, add #k360^@#
Check by calculator.
#x = 55.57 --> 2x = 111.14 --> sin (2x + 57.32) = sin (168.46) = 0.20#. Proved