sin (4x) = .5 what are the 4 solutions? How did you get them?

I saw your other post.

the way I do these is .....

treat 4x as your angle, ignore the fact it is a multiple of 4

by CAST rule
4x is in quadrants I and II since the sine is + in those quadrants
so 4x = 30° or 150°
(since sin30 = .5 and sin150 = .5)

if 4x = 30, then x = 30/4 or 7.5°
if 4x = 150, then x = 150/4 = 37.5°

now the period of sin 4x = 360°/4 = 90°
so by adding/subtracting any multiple of 90° we get other answers, that is
7.5+90= 97.5
37.5+90 = 127.5

97.5+90 = 187.5
127.5+90=217.5

187.5+90=277.5
217.5+90=307.5

adding another 90 will exceed the 360°

so this equation actually has 8 different answers

now recalling our previous question, I actually omitted 2 answers, there should have been 4.

I recall our angle was 2x, so the period would have been 360/2 = 180°
so by adding 180 to the 2 answers I gave you, will produce the other two remaining angles.

In general for both sin (kØ) and cos(kØ)
the period would be 360/k and there would be 2k solutions

Okay that makes sense. I knew i was overlooking something. Thanks for the help Reiny! it really helped me out

You are welcome

To find the four solutions to the equation sin(4x) = 0.5, we can use inverse trigonometric functions.

First, let's identify the range of the sine function. Since the sine function oscillates between -1 and 1, we need to find the values of x where sin(4x) equals 0.5 within one oscillation period.

To get the solutions, we can start by taking the inverse sine (also known as arcsin) of both sides of the equation:

arcsin(sin(4x)) = arcsin(0.5)

Since the sine and arcsin functions cancel each other out, we have:

4x = arcsin(0.5)

Now, we need to find the principal angle where sin(x) equals 0.5. Looking at the unit circle or referencing a trigonometric table, we find that sin(x) = 0.5 at two angles: 30 degrees (or π/6 radians) and 150 degrees (or 5π/6 radians).

However, we need to consider the periodicity of the sine function. Since sin(x) repeats every 360 degrees (or 2π radians), we add multiples of these values to find all possible solutions.

So, the four solutions for 4x are:

1) 4x = 30 degrees + 360 degrees * n, where n is an integer
2) 4x = 150 degrees + 360 degrees * n, where n is an integer

To find the actual values of x, we divide both sides of each equation by 4:

1) x = (30 degrees + 360 degrees * n) / 4, where n is an integer
2) x = (150 degrees + 360 degrees * n) / 4, where n is an integer

Now you can substitute different values of n to find the four solutions for x.