Answer:
x ≈ -1.97, x ≈ -0.17, x ≈ 1.43, x ≈ -1.05, and x ≈ 1.63
Explanation:
We are given the equation:
5 sin (3x + 0.1) + 2 = 0
Subtracting 2 from both sides, we get:
5 sin (3x + 0.1) = -2
Dividing by 5, we get:
sin (3x + 0.1) = -0.4
To solve for x, we need to find the angle whose sine is -0.4. We know that the sine function is negative in the third and fourth quadrants of the unit circle. Therefore, we need to find the reference angle whose sine is 0.4 and then add or subtract multiples of π to get angles in the third and fourth quadrants.
Using a calculator, we find that the reference angle whose sine is 0.4 is approximately 0.41 radians.
Therefore, we have:
sin θ = 0.4
θ ≈ 0.41 radians
To find the angles in the third and fourth quadrants, we subtract and add π, respectively. Therefore, we have:
3x + 0.1 = -0.41 + nπ (for some integer n)
or
3x + 0.1 = π - 0.41 + nπ (for some integer n)
Solving for x in each equation, we get:
3x = -0.51 + nπ or 3x = 0.69 + nπ
Dividing by 3, we get:
x = (-0.51/3) + (nπ/3) or x = (0.69/3) + (nπ/3)
Simplifying, we have:
x ≈ -0.17 + (nπ/3) or x ≈ 0.23 + (nπ/3)
Since we are given that -π/2
We need to find the values of n that make the solutions lie in this interval.
For the first equation, we have:
-π/2 < -0.17 + (nπ/3) < π
Adding π/2 to all parts of the inequality, we get:
-0.67 < (nπ/3) + π/2 < π/2
Multiplying by 3/π, we get:
-1.71 < n < 0.64
The integer values of n that satisfy this inequality are -1, 0, and 1.
Therefore, the solutions for x in the interval (-π/2, π) are:
x ≈ -0.17 - (π/3), x ≈ -0.17, x ≈ -0.17 + (π/3), x ≈ 0.23 - (π/3), and x ≈ 0.23 + (π/3)
Rounding each solution to 2 decimal places, we have:
x ≈ -1.97, x ≈ -0.17, x ≈ 1.63, x ≈ -1.05, and x ≈ 1.43
Therefore, the solutions to the given equation, in radians and rounded to 2 decimal places, are:
x ≈ -1.97, x ≈ -0.17, x ≈ 1.43, x ≈ -1.05, and x ≈ 1.63