Find the best solutions to your problems with the help of IDNLearn.com's experts. Find reliable solutions to your questions quickly and accurately with help from our dedicated community of experts.
Sagot :
Certainly! Let's solve the given trigonometric equation step by step:
[tex]\[ 9 \cos^2 x - 4 \cos x - 1 = 0 \][/tex]
1. Substitute [tex]\(\cos(x)\)[/tex] with [tex]\(u\)[/tex]:
Let [tex]\(u = \cos(x)\)[/tex]. Then the equation becomes a quadratic in [tex]\(u\)[/tex]:
[tex]\[ 9u^2 - 4u - 1 = 0 \][/tex]
2. Solve the quadratic equation:
The general form of a quadratic equation is [tex]\(au^2 + bu + c = 0\)[/tex]. For our equation, [tex]\(a = 9\)[/tex], [tex]\(b = -4\)[/tex], and [tex]\(c = -1\)[/tex].
3. Calculate the discriminant:
The discriminant of a quadratic equation [tex]\(au^2 + bu + c = 0\)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
Here, [tex]\( \Delta = (-4)^2 - 4 \cdot 9 \cdot (-1) = 16 + 36 = 52 \)[/tex].
4. Find the roots using the quadratic formula:
The quadratic formula states that the solutions for [tex]\(u\)[/tex] are given by:
[tex]\[ u = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Plugging in our values:
[tex]\[ u_{1} = \frac{-(-4) + \sqrt{52}}{2 \cdot 9} = \frac{4 + 2\sqrt{13}}{18} = \frac{2 + \sqrt{13}}{9} \approx 0.6228 \][/tex]
[tex]\[ u_{2} = \frac{-(-4) - \sqrt{52}}{2 \cdot 9} = \frac{4 - 2\sqrt{13}}{18} = \frac{2 - \sqrt{13}}{9} \approx -0.1784 \][/tex]
5. Convert [tex]\(u\)[/tex] back to [tex]\(x\)[/tex] using the arccosine function:
To find [tex]\(x\)[/tex], we take the arccosine of both [tex]\(u_{1}\)[/tex] and [tex]\(u_{2}\)[/tex]:
[tex]\[ x_{1} = \arccos(u_{1}) \approx \arccos(0.6228) \approx 0.8984 \, \text{(radians)} \][/tex]
[tex]\[ x_{2} = \arccos(u_{2}) \approx \arccos(-0.1784) \approx 1.7501 \, \text{(radians)} \][/tex]
6. Find additional solutions within the range [tex]\([0, 2\pi)\)[/tex]:
Since the cosine function is periodic and symmetric, we have additional solutions:
[tex]\[ x_{3} = 2\pi - x_{1} = 2\pi - 0.8984 \approx 5.3848 \, \text{(radians)} \][/tex]
[tex]\[ x_{4} = 2\pi - x_{2} = 2\pi - 1.7501 \approx 4.5330 \, \text{(radians)} \][/tex]
Hence, the detailed solutions for [tex]\(x\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex] are:
- [tex]\( x_{1} \approx 0.8984 \, \text{radians} \)[/tex]
- [tex]\( x_{2} \approx 1.7501 \, \text{radians} \)[/tex]
- [tex]\( x_{3} \approx 5.3848 \, \text{radians} \)[/tex]
- [tex]\( x_{4} \approx 4.5330 \, \text{radians} \)[/tex]
[tex]\[ 9 \cos^2 x - 4 \cos x - 1 = 0 \][/tex]
1. Substitute [tex]\(\cos(x)\)[/tex] with [tex]\(u\)[/tex]:
Let [tex]\(u = \cos(x)\)[/tex]. Then the equation becomes a quadratic in [tex]\(u\)[/tex]:
[tex]\[ 9u^2 - 4u - 1 = 0 \][/tex]
2. Solve the quadratic equation:
The general form of a quadratic equation is [tex]\(au^2 + bu + c = 0\)[/tex]. For our equation, [tex]\(a = 9\)[/tex], [tex]\(b = -4\)[/tex], and [tex]\(c = -1\)[/tex].
3. Calculate the discriminant:
The discriminant of a quadratic equation [tex]\(au^2 + bu + c = 0\)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
Here, [tex]\( \Delta = (-4)^2 - 4 \cdot 9 \cdot (-1) = 16 + 36 = 52 \)[/tex].
4. Find the roots using the quadratic formula:
The quadratic formula states that the solutions for [tex]\(u\)[/tex] are given by:
[tex]\[ u = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Plugging in our values:
[tex]\[ u_{1} = \frac{-(-4) + \sqrt{52}}{2 \cdot 9} = \frac{4 + 2\sqrt{13}}{18} = \frac{2 + \sqrt{13}}{9} \approx 0.6228 \][/tex]
[tex]\[ u_{2} = \frac{-(-4) - \sqrt{52}}{2 \cdot 9} = \frac{4 - 2\sqrt{13}}{18} = \frac{2 - \sqrt{13}}{9} \approx -0.1784 \][/tex]
5. Convert [tex]\(u\)[/tex] back to [tex]\(x\)[/tex] using the arccosine function:
To find [tex]\(x\)[/tex], we take the arccosine of both [tex]\(u_{1}\)[/tex] and [tex]\(u_{2}\)[/tex]:
[tex]\[ x_{1} = \arccos(u_{1}) \approx \arccos(0.6228) \approx 0.8984 \, \text{(radians)} \][/tex]
[tex]\[ x_{2} = \arccos(u_{2}) \approx \arccos(-0.1784) \approx 1.7501 \, \text{(radians)} \][/tex]
6. Find additional solutions within the range [tex]\([0, 2\pi)\)[/tex]:
Since the cosine function is periodic and symmetric, we have additional solutions:
[tex]\[ x_{3} = 2\pi - x_{1} = 2\pi - 0.8984 \approx 5.3848 \, \text{(radians)} \][/tex]
[tex]\[ x_{4} = 2\pi - x_{2} = 2\pi - 1.7501 \approx 4.5330 \, \text{(radians)} \][/tex]
Hence, the detailed solutions for [tex]\(x\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex] are:
- [tex]\( x_{1} \approx 0.8984 \, \text{radians} \)[/tex]
- [tex]\( x_{2} \approx 1.7501 \, \text{radians} \)[/tex]
- [tex]\( x_{3} \approx 5.3848 \, \text{radians} \)[/tex]
- [tex]\( x_{4} \approx 4.5330 \, \text{radians} \)[/tex]
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Find clear and concise answers at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.