find-solution-set-of-equation-cos-3x-cos-x-x-0-3pi-

Question Number 114272 by bemath last updated on 18/Sep/20

f i n d s o l u t i o n s e t o f e q u a t i o n

\cos 3 x = - \cos x, x \in (0, 3 π)

Commented by malwaan last updated on 18/Sep/20

(i) c o s 3 x = c o s (π - x)

\Rightarrow 3 x = \pm (π - x) + 2 k π

(a) 3 x = (π - x) + 2 k π

\Rightarrow x = π (\frac{1 + 2 k}{4})

(b) 3 x = - (π - x) + 2 k π

\Rightarrow x = π (\frac{2 k - 1}{2})

(i i) c o s 3 x = c o s (π + x)

\Rightarrow 3 x = \pm (π + x) + 2 k π

(a) 3 x = (π + x) + 2 k π

\Rightarrow x = π (\frac{1 + 2 k}{2})

(b) 3 x = - (π + x) + 2 k π

\Rightarrow x = π (\frac{2 k - 1}{4})

∴ s o l u t i o n s e t =

{π (\frac{2 k \pm 1}{2}); π (\frac{2 k \pm 1}{4})}

Commented by bemath last updated on 18/Sep/20

h a h a h a s a n t u y

Commented by abdullahquwatan last updated on 18/Sep/20

k e n a p a p a k e s a n t u y s i h ? ?

Answered by bemath last updated on 18/Sep/20

Answered by malwaan last updated on 18/Sep/20

x = {π (\frac{2 k \pm 1}{2}); π (\frac{2 k \pm 1}{4})}

x \in (0, 3)

k = 1 \Rightarrow x = \frac{π}{2}; \frac{3 π}{2}; \frac{π}{4}; \frac{3 π}{4} (< 3 π)

k = 2 \Rightarrow x = \frac{5 π}{2}; \frac{5 π}{4} (< 3 π)

k = 3 \Rightarrow x = \frac{7 π}{4} (< 3 π)

k = 4 \Rightarrow x = \frac{9 π}{4} (< 3 π)

k = 5 \Rightarrow x = \frac{11 π}{4} (3 π)

Answered by Dwaipayan Shikari last updated on 18/Sep/20

c o s 3 x + c o s x = 0

2 c o s 2 x c o s x = 0

c o s 2 x = 0

2 x = k π + \frac{π}{2}

x = \frac{k π}{2} + \frac{π}{4}

o r

x = k π + \frac{π}{2} (k \in Z)

S o l u t i o n s e t

x \in [\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}, \frac{9 π}{4}, \frac{11 π}{4}]

x \in [\frac{π}{2}, \frac{3 π}{2}, \frac{5 π}{2}]