Question

How many solutions does the equation cos(6x)=1/2 have on the interval (0, 2pi)?

Answer 1

Cos( 6 x ) = 1/2          
6 x = 60°  ⇒ x 1 = 10°
6 x = 300° ⇒ x 2 = 50°
6 x = 420° ⇒ x 3 = 70°
6 x  = 660° ⇒ x 4 = 110°
x 5 = 130°,  x 6 = 170°,  x 7 = 190° , x 8 = 230°, x 9 = 250°,
x 10 = 290°, x 11 = 310°, x 12 = 350°
Answer: there are 12 solutions on the interval ( 0 , 2π ).

Answer 2

Answer:

The number of possible solution of the trignometric equation are:

12

Step-by-step explanation:

We have to find the solution of the trignometric equation which is given as:

$\cos(6x)=\dfrac{1}{2}$

Now, the solution of the trignometric equation is the possible value of x such that the equation holds true.

Now we know that:

$\cos(\dfrac{\pi}{3})=\dfrac{1}{2}$

Also,

$\cos(2\pi-\dfrac{\pi}{3})=\dfrac{1}{2}\\\\i.e.\\\\\cos(\dfrac{5\pi}{3})=\dfrac{1}{3}$

We are given that:

0<x<2π.

this means that:

0<6x<12π.

Now, we have the solution as:

$6x=\dfrac{\pi}{3}\\\\\\i.e\\\\x=\dfrac{\pi}{18}$

and:

$6x=\dfrac{5\pi}{3}\\\\x=\dfrac{5\pi}{18}$

$6x=2\pi+\dfrac{\pi}{3}=\dfrac{7\pi}{3}\\\\6x=4\pi-\dfrac{\pi}{3}=\dfrac{11\pi}{3}\\\\6x=4\pi+\dfrac{\pi}{3}=\dfrac{13\pi}{3}\\\\6x=6\pi-\dfrac{\pi}{3}=\dfrac{17\pi}{3}\\\\6x=6\pi+\dfrac{\pi}{3}=\dfrac{19\pi}{3}\\\\6x=8\pi-\dfrac{\pi}{3}=\dfrac{23\pi}{3}\\\\6x=8\pi+\dfrac{\pi}{3}=\dfrac{25\pi}{3}\\\\6x=10\pi-\dfrac{\pi}{3}=\dfrac{29\pi}{3}\\\\6x=10\pi+\dfrac{\pi}{3}=\dfrac{31\pi}{3}\\\\6x=12\pi-\dfrac{\pi}{3}=\dfrac{35\pi}{3}$

Hence, the number of possible solution of the trignometric equation are:

12