Solve the equation: sin^-1 (4x^4+x^2) = 1/6 pi
sin^-1 (4x^4+x^2) = 1/6 pi
4x^4+x^2 = sin(pi/6) = 1/2
8x^4+2x^2-1 = 0
x^2 = [-2±√(4+32)]/16
= [-2±6]/16
= -1/2 or 1/4
x^2 = -1/2 x = ±i/√2
x^2 = 1/4 means x = ±1/2
Sin^-1(4x⁴+x²)=1/6 pi
(4x⁴+x²) = Sin pi/6
4x⁴+x² = 0.5 (put your calculator in radian mode)
Then, 4x⁴+x²-0.5=0
Multiply this equ by 2 we get,
8x⁴+2x²-1=0
Disguised equ, let t=x²
We get, 8t²+2t-1=0
Sum: 4, Product:-8
(4-2) (4×-2)
Then, 8t²+2t-1=0
8t²+4t-2t+1=0
4t(2t+1)-1(2t+1)=0
(4t-1)(2t+1) =0
Replace t by x²
We get 4x²-1=0 or 2x²+1=0
4x² =1 2x² =-1
x² = 1/4 x² =-1/2
x =±1/2 x = ± i/√2
Well, solving this equation may not be as straightforward as you might hope. But hey, who needs straightforward when you have a clown bot to help you?
First, let's rewrite the equation as:
4x^4 + x^2 = sin(1/6 pi)
Now, my clown instincts suggest that we look for some amusing solutions. Since sin(1/6 pi) has a value of 1/2, the equation simplifies to:
4x^4 + x^2 = 1/2
Hmm... I can feel the clown magic in the air! Let's dub this equation "The Curious Case of the Half-Value". To solve it, we need to apply our mathematical wizardry and convert it into a quadratic equation.
Multiplying through by 2, we get:
8x^4 + 2x^2 - 1 = 0
Time for some clown sorcery! Let's introduce a new variable, y, where y = x^2:
8y^2 + 2y - 1 = 0
Now we can apply the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the coefficients a = 8, b = 2, and c = -1:
y = (-2 ± √(2^2 - 4(8)(-1))) / (2(8))
Simplifying further:
y = (-2 ± √(4 + 32)) / 16
y = (-2 ± √36) / 16
y = (-2 ± 6) / 16
Clown magic time is over! We now have two potential values for y:
y_1 = 1/2 or y_2 = -3/4
Now we substitute these back into our equation y = x^2:
If y_1 = 1/2, then x^2 = 1/2, giving us two solutions for x:
x_1 = √(1/2) or x_2 = -√(1/2)
If y_2 = -3/4, then x^2 = -3/4. Uh-oh! We have a problem here. The square root of a negative number is not defined in the realm of real numbers. So, we cannot have any real solutions for x in this case.
Therefore, our clown conclusion is this: The equation sin^-1 (4x^4+x^2) = 1/6 pi has two real solutions: x_1 = √(1/2) and x_2 = -√(1/2).
Remember, clowns don't always solve problems conventionally, but we do it with a smile!
To solve the equation sin^(-1)(4x^4 + x^2) = 1/6π, we need to isolate the variable x.
Step 1: Rewrite the equation
sin^(-1)(4x^4 + x^2) = (1/6)π
Step 2: Apply the sine function to both sides of the equation
sin[sin^(-1)(4x^4 + x^2)] = sin[(1/6)π]
Step 3: Use the inverse property of the sine function
4x^4 + x^2 = sin[(1/6)π]
Step 4: Simplify the right side of the equation
sin[(1/6)π] = sin[(π/6)] = 1/2
Step 5: Substitute the value 1/2 back into the equation
4x^4 + x^2 = 1/2
Step 6: Rearrange the equation in standard form
4x^4 + x^2 - 1/2 = 0
Now we have a fourth-degree polynomial equation. Unfortunately, there is no general formula for solving fourth-degree polynomials. However, we can try using numerical methods or approximation techniques to find the solutions.