To solve the trigonometric equation [tex]\(2 \sin^2(x) - \sin(x) - 1 = 0\)[/tex] for [tex]\(0^\circ \leq x \leq 90^\circ\)[/tex], follow the steps below in the correct order:
1. Recognize the equation as a quadratic equation in terms of [tex]\(\sin(x)\)[/tex]:
[tex]\(2 (\sin(x))^2 - \sin(x) - 1 = 0\)[/tex].
2. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in the quadratic equation [tex]\(a (\sin(x))^2 + b \sin(x) + c = 0\)[/tex]:
Here, [tex]\(a = 2\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -1\)[/tex].
3. Calculate the discriminant [tex]\(\Delta = b^2 - 4ac\)[/tex]:
[tex]\(\Delta = (-1)^2 - 4 \cdot 2 \cdot (-1) = 1 + 8 = 9\)[/tex].
4. Take the square root of the discriminant:
[tex]\(\sqrt{\Delta} = \sqrt{9} = 3.0\)[/tex].
5. Find the two roots using the quadratic formula [tex]\(\sin(x) = \frac{-b \pm \sqrt{\Delta}}{2a}\)[/tex]:
[tex]\[\sin(x) = \frac{-(-1) \pm 3.0}{2 \cdot 2} = \frac{1 \pm 3.0}{4}\][/tex]
6. Compute the values for the two roots:
[tex]\[\sin(x_1) = \frac{1 + 3.0}{4} = 1.0\][/tex]
[tex]\[\sin(x_2) = \frac{1 - 3.0}{4} = -0.5\][/tex]
7. Determine the valid solutions for [tex]\(x\)[/tex] within the range [tex]\(0^\circ \leq x \leq 90^\circ\)[/tex]:
[tex]\(\sin(x) = 1.0\)[/tex] corresponds to [tex]\(x = 90^\circ\)[/tex].
[tex]\(\sin(x) = -0.5\)[/tex] is not within the range [tex]\([-1, 1]\)[/tex] for [tex]\(\sin(x)\)[/tex] in the specified quadrant.
Therefore, the solution is: [tex]\(x = 90^\circ\)[/tex].
The valid solutions within the range [tex]\(0^\circ \leq x \leq 90^\circ\)[/tex] is:
[tex]\[
[90.0]
\][/tex]
