SAT Math Problem (9)

Question

For how many values of x between 0 and 2π does sin[3x] = 1/2?

Answer 1

The "3x" is a distraction (typical of SAT questions).

sin θ = 0.2 for only two values, θ = π/6, or θ = 5π/6.

Therefore x must be either π/18 or 5π/18.

 

Answer 2

sin(3x) = 1/2

 

3x = π/6 + k(2π) or 5π/6 + k(2π), where k = 0, ±1, ±2, ...

 

So, x = π/18 +k(2π/3) or 5π/18 + k(2π/3)

 

The values of x between 0 and 2π are obtained by substituting in k = 0, 1, and 2.

 

They are:  π/18, 5π/18, 13π/18, 17π/18, 25π/18, and 29π/18

 

So, there are a total of 6 values.  

 

Answer 3

Lets use the addition angle identity to get an accurate result.

 

sin(2x + x) = 1/2

 

sin(2x)cos(x) + cos(2x)sin(x) = 1/2

 

 

Now, we apply some trig identities on the double-angle identity terms.

 

(2sin(x)cos(x))cos(x) + (1 - 2sin²(x))sin(x) = 1/2

 

2sin(x)cos²(x) + sin(x) - 2sin³(x) = 1/2

 

2sin(x)(1 - sin²(x)) + sin(x) - 2sin³(x) = 1/2

 

2sin(x) - 2sin³(x) + sin(x) - 2sin³(x) = 1/2

 

3sin(x) - 4sin³(x) = 1/2

 

 

Move all terms to the right side of the equation.

 

0 = 4sin³(x) - 3sin(x) + (1/2)

 

 

Multiply both sides of the equation by 2.

 

0 = 8sin³(x) - 6sin(x) + 1

 

 

Next, use synthetic division to find your solutions.  Remember that sine is only positive in the 1st and 2nd quadrants.