SOLUTION: For the equation sin(8 x) = 0.97 find the smallest solution, the largest solution and the number of solutions for x in the interval 0 ≤ x ≤ 2π. The first t

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Question 969721: For the equation
sin(8 x) = 0.97
find the smallest solution, the largest solution and the number of solutions for x in the interval 0 ≤ x ≤ 2π.
The first two answers should be given accurate to two decimal places.
My answers (not correct) were:
0.12
5.77
16
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
you have sin(8x) = .97
the period of 8x is equal to 2pi/8.
there will be 8 repetititions of a full sine wave sycle in a standard 0 to 2pi interval.

use your calculator to find that sin(8x) = .97 leads to an angle of 8x = 1.325230809.

since sine is positive in the first and second quadrant, than you need to get the angle in the second quadrant as well.

that angle sill be pi - 1.325230809 = 1.816361844.

those angles are 8x.

you want angles of x.

divide those angles by 8 and you will get:

x1 = .1656538512
x2 = .2270452305

those angles will repeat every full cycle of 2pi/8.

your solution is therefore equal to:

x1 = .1656538512 plus or minus k * 2pi/8.
x2 = .2270452305 plus or minus k * 2pi/8.

your smallest angle in the interval from 0 to 2pi will be x1 = .1656538512.

your largest angle in the interval from 0 to 2pi will be x2 = .2270452305 + 7*2pi/8 = 5.724832374.

the graph below shows the function of y = sin(8x) in the interval from 0 to 2pi.
the smallest value of x and the largest value of x where sin(x) = .97 are marked.

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