Solutions of $\sin(\cos x)=1$

Algebra Level 3

Find the number of solutions of $\sin(\cos x)=1$ in the interval $x \in [0,4\pi]$ .

2 0 5 4

3 solutions

Harri Bell-Thomas
Oct 12, 2014

The graph of y = sin(cosx) will have maximums at less than 1, therefore meaning there will be no solutions.

Even the sine is equal to one only when the angle is an odd multiple of $π/2$ . That means $cos x$ has to be equal to an odd multiple of $π/2$ . Even considering the lowest multiple ( $1$ ), $π/2$ is nearly equal to $1.57$ . But the range of cosine is between $1$ and $-1$ . Therefore, there is no such value of $x$ which satisfies the equation.

Arkajyoti Banerjee - 5 years, 1 month ago

Chew-Seong Cheong
Mar 14, 2017

Since $\cos x \in [-1,1]$ , $\sin (\cos x) \in [\sin(-1), \sin (1)]$ . Since $\sin (1) < 1$ , $\implies \sin (\cos x) < 1$ and the equation has $\boxed{0}$ solution.

Akeel Howell
Mar 25, 2017

If $\sin{y} = 1$ then $y = a\dfrac{\pi}{2}$ where $a = 4n+1 \space \space \forall \space \space n \in \mathbb{Z}$ .

But $y = \cos{x}$ and $-1 \leq \cos{x} \leq 1 \space \space \forall \space \space x \in \mathbb{R}$ . Thus, there are no real solutions to $\sin(\cos{x}) = 1$ .

Solutions of sin ⁡ ( cos ⁡ x ) = 1 \sin(\cos x)=1 sin ( cos x ) = 1

3 solutions

0 pending reports

Solutions of $\sin(\cos x)=1$