What is the maximum value of \(8\sin\theta - 4\sin^2\theta\)?
Solving for the Maximum Value of \(8\sin\theta - 4\sin^2\theta\)
The problem asks for the maximum value of the expression \(8\sin\theta - 4\sin^2\theta\). This expression involves the sine function, \(\sin\theta\). We can simplify this problem by making a substitution and analyzing the resulting algebraic expression.
Algebraic Approach Using Substitution
1. Define a Variable
Let \(x = \sin\theta\). Since the range of the sine function is from -1 to 1, we know that \(x\) must be in the interval \([-1, 1]\).
2. Rewrite the Expression
Substitute \(x\) for \(\sin\theta\) in the given expression:
\( 8\sin\theta - 4\sin^2\theta = 8x - 4x^2 \)
Now, we need to find the maximum value of the quadratic function \(f(x) = -4x^2 + 8x\) within the domain \(-1 \le x \le 1\).
3. Analyze the Quadratic Function
The function \(f(x) = -4x^2 + 8x\) represents a parabola. Since the coefficient of the \(x^2\) term (which is -4) is negative, the parabola opens downwards, meaning it has a maximum point at its vertex.
4. Find the Vertex
The x-coordinate of the vertex of a parabola in the form \(ax^2 + bx + c\) is given by the formula \(x = -\frac{b}{2a}\).
In our function, \(f(x) = -4x^2 + 8x\), we have \(a = -4\) and \(b = 8\). Plugging these values into the formula:
\( x_{vertex} = -\frac{8}{2(-4)} = -\frac{8}{-8} = 1 \)
5. Check Domain Validity
The vertex occurs at \(x = 1\). We need to check if this value is within the allowed domain for \(x\), which is \([-1, 1]\). Fortunately, \(x=1\) is within this domain (it's the right endpoint).
6. Calculate the Maximum Value
Since the vertex is within the domain and the parabola opens downwards, the maximum value of the function \(f(x)\) occurs at the vertex, \(x=1\). Substitute \(x=1\) back into the function:
\( f(1) = -4(1)^2 + 8(1) \)
\( f(1) = -4(1) + 8 \)
\( f(1) = -4 + 8 \)
\( f(1) = 4 \)
This maximum value occurs when \(\sin\theta = 1\), which is possible for values like \(\theta = \frac{\pi}{2}\) radians (or 90 degrees).
Conclusion
The maximum value of the expression \(8\sin\theta - 4\sin^2\theta\) is 4.
Similar Questions
What is the minimum value of \(\left( \frac{a^2 + 3a + 1}{a} \right) \left( \frac{b^2 + 3b + 1}{b} \right)\) for \(a, b > 0\)?
- A real number \(x\) is such that the sum of the number and four times its square is the least. What is that number?
What is the minimum value of \(\frac{(a^8+a^4+1)(b^8+b^4+1)}{a^4b^4}\) where \(a > 0, b > 0\) ?
Important Questions from Maxima and Minima
Four small squares of side x are cut out of a square of side 12 cm to make a tray by folding the edges. What is the value of x so that the tray has the maximum volume?
The maximum value of 5 + 20x - 4x², when x is a real number is
- The minimum value of the expression max {$p^2 - 2p + 4, -p^2 + 2p - 4$} in the range $0 \le p \le 1$ is:
- If $a^2 + b^2 + c^2 + d^2 = 1$, what will be the maximum value of the product abcd?
If 3 ≤ x ≤ 10 and 5 ≤ y ≤ 15 , then maximum value of \(\left(\frac{x}{y}\right)\) is-
