How to solve algebraiclly?

for the time being, think of sin(10x) as just a number u.

30u - 16 = 6

30u = 22

u = 22/30 = 11/15

so

sin(10x) = 11/15

to find x, we need to apply the arcsin() function, which is the inverse function of sin.

so 10x = arcsin(sin(10x)) = arcsin(11/15)

now, theis will only gives us back angles between -π/2 and π/2 (-180 degrees or 180 degrees)

and since sin is periodic, other answers are possible.

arcsin(11/15) ~ 0.823 radians or 47.17 degrees. this is 10x.

so x ~ 0.0823 radians or 4.717 degrees (this is the principal value of x).

or course other values of x will still give us the same answer:

since sin(y) = sin(y + 2kπ), x + πk/5 will also give us the same result (in radians, in degrees this would be x + 36k) for ANY integer (negative or positive) k.

since sin(π/2 - y) = sin(π/2 + y), writing 10x = π/2 - (π/2 - 10x) we see we will also get the same answer for 10x2 = π/2 + (π/2 - 10x) = π - 10x

or x2 = π/10 - x (or in degrees, 18 - x).

finally, we will also get the same answer for any k with πk/5 + x2

(= (2k+1)π/10 - x, and in degrees 36k + x2 = 36k + 18 - x).

as you can see, a complete answer is a bit complicated, because of the fact that sin is periodic.

often, a restriction is put on the possible values of x, such as:

0 ≤ x < 2π (or in degrees, 0 ≤ x < 360), for example.