Solve the equation: sin^-1 (4x^4+x^2) = 1/6 pi
sin^-1 (4x^4+x^2) = 1/6 pi
take sine of both sides:
4x^4+x^2 = 1/2
8x^4+2x^1-1 = 0
(4x^2-1)(2x^2+1) = 0
...
To solve the equation sin^(-1)(4x^4 + x^2) = (1/6)π, we need to find the value of x.
Step 1: Rewrite the equation using the inverse sine function notation:
arcsin(4x^4 + x^2) = (1/6)π
Step 2: Take the sine of both sides to cancel out the arcsine:
sin(arcsin(4x^4 + x^2)) = sin((1/6)π)
Step 3: Simplify the left side using the inverse sine identity:
4x^4 + x^2 = sin((1/6)π)
Step 4: Simplify the right side by evaluating the sine of (1/6)π:
4x^4 + x^2 = sin(π/6)
4x^4 + x^2 = 1/2
Step 5: Rearrange the equation:
4x^4 + x^2 - 1/2 = 0
Step 6: Substitute a variable to simplify the equation:
Let's substitute a = x^2, so our equation becomes:
4a^2 + a - 1/2 = 0
Step 7: Solve the quadratic equation:
To solve this equation, we can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula.
The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a).
For our equation, a = 4, b = 1, and c = -1/2.
Plugging these values into the quadratic formula, we get:
x = (-(1) ± √((1)^2 - 4(4)(-1/2))) / (2(4))
Simplifying further:
x = (-1 ± √(1 + 8)) / 8
x = (-1 ± √9) / 8
x = (-1 ± 3) / 8
So, we have two possible solutions:
x1 = (3 - 1) / 8 = 2/8 = 1/4
x2 = (-3 - 1) / 8 = -4/8 = -1/2
Hence, the solutions to the equation sin^(-1)(4x^4 + x^2) = (1/6)π are x = 1/4 and x = -1/2.
To solve the equation sin^(-1) (4x^4 + x^2) = 1/6π, we need to isolate the variable x. Here's how you can do that:
Step 1: Rewrite the equation in its equivalent form using the inverse sine function sin^(-1):
4x^4 + x^2 = sin(1/6π)
Step 2: Take the sine of both sides to eliminate the inverse sine function:
sin(4x^4 + x^2) = sin(sin(1/6π))
Step 3: Simplify the right side of the equation:
sin(4x^4 + x^2) = sin(π/6)
Step 4: Use the fact that sin(x) = sin(π - x) to rewrite the equation:
sin(4x^4 + x^2) = sin(π - π/6)
Step 5: Simplify the right side of the equation:
sin(4x^4 + x^2) = sin(5π/6)
Step 6: Now, we need to find the values of x that satisfy the equation sin(4x^4 + x^2) = sin(5π/6). To do this, we'll set the argument of the sine function equal to the argument of the other sine function:
4x^4 + x^2 = 5π/6
Step 7: Rearrange the equation:
4x^4 + x^2 - 5π/6 = 0
At this point, the equation is a quadratic equation in terms of x^2. You can use various methods to solve this equation, such as factoring, completing the square, or using the quadratic formula.