Question

Number of solution(s) of the equation $\cos (2x+\frac{\pi }{3})=1$ in $[0,3\pi ]$ is/are

• A. 3.00
• B. 03
• C. 3
• D. 3.0

Answer 1

Answer: (A), (B), (C), (D)

Number of solutions of the equation $\cos (2x+\frac{\pi }{3})=1$ in $[0,3\pi ]$ are same as the points of intersection of the graphs $y=\cos (2x+\frac{\pi }{3})$ and $y=1$.

In $y=\cos (2x+\frac{\pi }{3})$ fundamental function involved is $\cos x$ , whose graph is given by

Apply the stretch transformation by 2 units in $y=\cos x$ to get $y=\cos 2x$ as shown below
Now apply the horizontal shift by $\frac{\pi }{6}$ to get $y=\cos (2x+\frac{\pi }{3})$ as shown below
Now let us calculate the number of point of intersections of $y=1$ and $y=\cos (2x+\frac{\pi }{3})$ in $[0,3\pi ]$ from the graph
Clearly in $[0,3\pi ]$ , there are 3 points of intersection.
Hence given equation have 3 solutions.