approximate the solutions (to three decimal places) of the given equation in the interval
square bracket 0, 2π)
sin 2x + 1.5 cos x = 0
"square bracket"? Is your keyboard defective, so you can't just type "[" ?
Even more confusing is that you didn't use what I gave you at
http://www.jiskha.com/display.cgi?id=1468255629
To approximate the solutions to the equation sin 2x + 1.5 cos x = 0 in the interval [0, 2π), we can follow these steps:
Step 1: Rearrange the equation:
sin 2x + 1.5 cos x = 0
2 sin x cos x + 1.5 cos x = 0
cos x (2 sin x + 1.5) = 0
Step 2: Solve for cos x = 0:
We have two solutions for cos x = 0 in the given interval, which are:
x = π/2
x = 3π/2
Step 3: Solve for 2 sin x + 1.5 = 0:
2 sin x + 1.5 = 0
2 sin x = -1.5
sin x = -1.5/2
sin x = -0.75
To find the solutions for sin x = -0.75 in the interval [0, 2π), we can use the inverse sine function:
Step 4: Solve for arcsin (-0.75) using a calculator:
arcsin (-0.75) ≈ -0.8481
We have one solution for sin x = -0.75 in the interval [0, 2π), which is:
x ≈ -0.8481
The approximate solutions to the given equation in the interval [0, 2π) are:
x = π/2
x = 3π/2
x ≈ -0.8481
To approximate the solutions to the given equation, we can use the following steps:
1. Rewrite the equation in terms of a single trigonometric function using the identities sin(2x) = 2sin(x)cos(x) and cos(x) = √(1 - sin^2(x)).
The equation sin(2x) + 1.5cos(x) = 0 becomes:
2sin(x)cos(x) + 1.5√(1 - sin^2(x)) = 0.
2. Let's simplify the equation further by making a substitution. Let y = sin(x). The equation now becomes:
2y√(1 - y^2) + 1.5√(1 - y^2) = 0.
3. Now, we can combine the two terms with √(1 - y^2) as a common factor:
√(1 - y^2)(2y + 1.5) = 0.
4. Set each factor equal to zero:
√(1 - y^2) = 0, implying that 1 - y^2 = 0.
Solving 1 - y^2 = 0 for y gives us y = ±1.
Also, 2y + 1.5 = 0, which implies that y = -0.75.
5. Now, we need to find the corresponding values of x for each y value we obtained.
For y = ±1, we know that sin(x) = ±1. The solutions in the interval [0, 2π) are x = π/2 and x = 3π/2.
For y = -0.75, we need to find the angles with sin(x) = -0.75. Using a calculator or trigonometric identity, we can find that x ≈ 4.188 and x ≈ 5.433.
6. Therefore, the approximate solutions (to three decimal places) of the given equation in the interval [0, 2π) are:
x ≈ π/2, 3π/2, 4.188, and 5.433.