Trig question; find all solutions for the given equation: 2sin3x=1?

Question

My book says the answer is x=10º + 120ºn or x=50º + 120ºn and I have no idea how to get that answer. Any help is greatly appreciated.

Answer 1

I will try to answer, because it's been a while since I did this.

The equation is: 2 sin 3x = 1

Dividing both sides by 2, the equation becomes : sin 3x = .5

The question now is what are the values of the angle whose sine values are equal to 0.5. There are 2 values:

a. 30 degrees
b. 150 degrees

But since the values of the sine repeat for every turn or after 360 degrees, we have to revise the answers to the following:

a. 30 degrees + 360 degrees x n
b. 150 degrees + 360 degrees x n

where n is the number of turns.

Therefore, our equation becomes:

3x = arcsin (0.5) = 30 degrees + 360 degrees x n

Therefore, dividing both sides by 3:

x = 10 degrees + 120 degrees x n.


For the other answer:

3x = arcsin (0.5) = 150 degrees + 360 degrees x n

Therefore : x = 50 degrees + 120 degrees x n.

Answer 2

2sin(3x)=1 => sin(3x) = 1/2, so to continue solving we need to find the angles where the sine of the angle is 1/2. Sin(30) is 1/2, so 180-30 = 150 will also have sine 1/2. After that, we need to add 360 degrees to either of these angles to make one circuit of the unit circle and get back to the angles with sine 1/2. So, all angles with sine 1/2 can be written on the form 30 + 360*n or 150+360*n where n is a whole number.

So, 3x = 30 + 360*n or 150+360*n , and dividing through with 3 gives you the answer you're looking for.

Answer 3

You could always draw a sketch. Anyway, rearranging, sin (3x) = 1/2. The first solution is that 3x= 30 deg, so x= 10 deg. If you follow the function sin(u) where u=3x along, the next point at which sin(u)=1/2 is at 150 deg, or 3x=50 deg. At 360 deg, sin(u)=0, and we repeat what happens from 0 to 360 deg. So the next point is at u=390 deg, or x=130, followed by u=510, or x=170. This is the result for n=1. The next repetition will give us x=250 and 290, and so on.

Answer 4

If you divide both sides by 2.. you get
sin3x = 1/2

ok... let's say 3x is y...
At what angles is sin(y) equal to 1/2?

30º, 150º and so on... right?
because the angles continue like a cycle...
30º + 360ºn and 150º + 360ºn would have been the answers.

however, remember we set y equal to 3x...
3x = 30º + 360ºn , 3x = 150º + 360ºn

to get x... we have to divide the answers by 3.
x = 10º + 120ºn and 50º + 120ºn

which are the given answers...

hope this helps...

Answer 5

Well to solve this first divide both sides by 2 to get:

sin3x = 1/2

Then multiply both sides by arcsin to get:

3x = arcsin(1/2)

Where from the rules of arcsin we know that the range for this function is from -pi/2 to pi/2 therefore limiting its answers away quadrant 2 and 3. You also see it is to 1/2 which is a 30, 60, 90 triangle and see that it is sin 1/2 so automatically know that it is 30 degrees (pi/6). SO then you get:

3x = pi/6

Where you then divide by 3 on both sides to get:

x = pi/18, which is the 10 degrees

Answer 6

csc(?) = 2 is called a million/sin(?) = 2. for this reason, sin(?) = a million/2 = 30°. also, it ability sin is positive, that would want to both be interior the I or II quadrant. The addition of 360° in basic terms shows one revolution. the answer is ? = 30° + 360°n or ? = one hundred and fifty° + 360°n, the position n is an integer. 30° falls in I quadrant and one hundred and fifty° falls in II quadrant.

Answer 7

2sin3x=1
sin 3x = 1/2 = 0.5

I know sin (30º) = 0.5, so
also sin (30º+ 360nº) = 0.5,
or

3x = sin^-1 (0.5) = 30º + 360nº
x = ( 30º + 360nº ) / 3
= 10º +120nº
for n = . + or - 1, 2, 3...

Answer 8

use the inverse processes