Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
[tex]2cos^2x-5cos+2=0\\\\cosx=t\in < -1;\ 1 >\\\\2t^2-5t+2=0\\\\a=2;\ b=-5;\ c=2\\\\\Delta=b^2-4ac;\ \Delta=(-5)^2-4\cdot2\cdot2=25-16=9\\\\t_1=\frac{-b-\sqrt\Delta}{2a};\ t_2=\frac{-b+\sqrt\Delta}{2a}\\\\t_1=\frac{5-\sqrt9}{2\cdot2}=\frac{5-3}{4}=\frac{2}{4}=\frac{1}{2}\in < -1;\ 1 >\\\\t_2=\frac{5+\sqrt9}{2\cdot2}=\frac{5+3}{4}=\frac{8}{4}=2\notin < -1;\ 1 >[/tex]
[tex]cosx=\frac{1}{2}\to x=\frac{\pi}{3}+2k\pi\ \vee\ x=-\frac{\pi}{3}+2k\pi\ \ \ (k\in\mathbb{Z})[/tex]
[tex]cosx=\frac{1}{2}\to x=\frac{\pi}{3}+2k\pi\ \vee\ x=-\frac{\pi}{3}+2k\pi\ \ \ (k\in\mathbb{Z})[/tex]
Answer:
[tex] x = \frac{\pi}{3} + 2n\pi \\ or\\ x=\frac{-\pi}{3} + 2n\pi [/tex]
where "n" is an integer that belongs to Z.
Explanation:
The equation given is:
2cos²(x) - 5cos(x) + 2 = 0
To factor this equation, we will use the quadratic formula shown in the attached image.
From the given equation:
a = 2
b = -5
c = 2
This means that:
either cos(x) = [tex] \frac{5+\sqrt{(-5)^2-4(2)(2)}}{2(2)} = 2 [/tex] .......> This solution is rejected as the value of the cosine function lies between -1 and 1 only.
or cos(x) = [tex] \frac{5-\sqrt{(-5)^2-4(2)(2)}}{2(2)} = 0.5 [/tex] ......> This solution is accepted as it lies within -1 and 1
Now, using the inverse of the cosine, we can find that:
[tex] x = \frac{\pi}{3} + 2n\pi \\ or\\ x=\frac{-\pi}{3} + 2n\pi [/tex]
where "n" is an integer that belongs to Z.
Hope this helps :)
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.