SOLUTION: How can i solve these trigonometric functions? Solve for x∈[0,2π]: {{{ sin(3x)=-1/sqrt(2) }}} also if you've got time Solve for x∈[0,2π]: {{{ sin(

Algebra ->  Trigonometry-basics -> SOLUTION: How can i solve these trigonometric functions? Solve for x∈[0,2π]: {{{ sin(3x)=-1/sqrt(2) }}} also if you've got time Solve for x∈[0,2π]: {{{ sin(      Log On


   

Question 1042988: How can i solve these trigonometric functions?
Solve for x∈[0,2π]:
+sin%283x%29=-1%2Fsqrt%282%29+
also if you've got time
Solve for x∈[0,2π]:
+sin%282x%29=-1%2F2+
I don't get how my answers are wrong after i solve it..
Thanks!!
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(53763) About Me  (Show Source):
You can put this solution on YOUR website!
How can i solve these trigonometric functions? 

1.  Solve for x∈[0,2π]:  +sin%283x%29=-1%2Fsqrt%282%29+

    +sin%283x%29 = -1%2Fsqrt%282%29  --->  3x = 5pi%2F4  and/or  3x = 7pi%2F4   --->   x = 3pi%2F12   and/or  x = 7pi%2F12



2.  Solve for x∈[0,2π]:  +sin%282x%29=-1%2F2+

    +sin%282x%29 = -1%2F2  --->  2x = 7pi%2F6  and/or  2x = 11pi%2F6   --->   x = 7pi%2F12   and/or  x = 11pi%2F12


Answer by Edwin McCravy(20081) About Me  (Show Source):
You can put this solution on YOUR website!
The good lady did not get all the solutions
between 0 and 2pi.  There are 6 solutions for
the first one and 4 solutions for the second one.

How can i solve these trigonometric functions?
Solve for x∈[0,2π]:
sin%283x%29=-1%2Fsqrt%282%29
The sine is negative in QIII and QIV

Rationalize the denominator of -1%2Fsqrt%282%29
and get -sqrt%282%29%2F2 and from your knowledge
of special angles or from the unit circle,
the solutions for 3x are 225° and 315° or
since you are using radians, 5pi%2F4 and 7pi%2F4.

However, the left side is sin(3x), not sin(x).

That means we must get 3 times as many answers for x
as we would get for 3x.

To do that we add 2pi 3-1 = 2 times to each of those
answers for 3x in order to get all non-negative answers for 
x less than 2pi:

So 

1.  we have to add 2pi and 4pi to the first answer5pi%2F4:  



which we get LCDs and simplify as



and simplify further as



Then divide through by 3 (multiply through by 1%2F3 and get:



or



Also,

2.  we have to add 2pi and 4pi to the second answer7pi%2F4:  



We get the LCD:



and simplify further as



Then divide through by 3 (multiply through by 1%2F3) and get:



or



All the 6 solutions in order of magnitude are



------------------------

Solve for x∈[0,2π]:

sin%282x%29=-1%2F2

As before, the sine is negative in QIII and QIV.

From your knowledge of special angles or from the unit 
circle, the solutions for sin(2x) are 210° and 330° or
since you are using radians, 7pi%2F6 and 11pi%2F6.

However, the left side is sin(2x), not sin(x).

That means we must get 2 times as many answers for x
as we would get for 2x.

To do that we add 2pi 2-1 = 1 time to each of those
answers for 2x in order to get all non-negative answers for 
x less than 2pi:

So 

1.  we have to add 2pi to the first answer7pi%2F6:  



We get the LCD:



and simplify further as

matrix%281%2C7%2C2x%2C%22%22%2C%22%22=%22%22%2C%22%22%2C7pi%2F6%2C%22%2C%22%2C19pi%2F6%29

Then divide through by 2 (multiply through by 1%2F2) and get:



or

matrix%281%2C7%2Cx%2C%22%22%2C%22%22=%22%22%2C%22%22%2C7pi%2F12%2C%22%2C%22%2C19pi%2F12%29

Also,

2.  we have to add 2pi to the second answer 11pi%2F6:  



which we get LCDs and simplify as



and simplify further as

matrix%281%2C7%2C2x%2C%22%22%2C%22%22=%22%22%2C%22%22%2C11pi%2F6%2C%22%2C%22%2C23pi%2F6%29

Then divide through by 2 (multiply through by 1%2F2 and get:



or

matrix%281%2C7%2Cx%2C%22%22%2C%22%22=%22%22%2C%22%22%2C11pi%2F12%2C%22%2C%22%2C23pi%2F12%29


All the 4 solutions in order of magnitude are

 

Edwin