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Find general solution
cos x-1=-cos x

User Gamalier
by
8.0k points

2 Answers

12 votes

Answer:

The general solution to
\cos(x - 1) = -\cos(x) is
x = (k + (1/2))\, \pi + (1/2) for all integer
k \in \mathbb{Z}.

Explanation:

Given:


\cos(x - 1) = -\cos(x).

Rearrange to obtain:


\cos(x - 1) + \cos(x) = 0.

By sum-of-angle identity for cosines:


\cos(a + b) = \cos(a) \,\cos(b) - \sin(a) \, \sin(b).

Since
\sin(-b) = -\sin(b), the following is also an identity:


\begin{aligned}\cos(a - b) &= \cos(a) \,\cos(b) - \sin(a) \, \sin(-b) \\ &= \cos(a)\, \cos(b) - \sin(a) \, (-\sin(b)) \\ &= \cos(a) \, \cos(b) + \sin(a)\, \sin(b)\end{aligned}.

Add these two identities together to obtain the sum-to-product formula:


\begin{aligned}& \cos(a + b) \\ &+ \cos(a - b)\\ &= \cos(a) \, \cos(b) - \sin(a)\, \sin(b) \\ &\quad + \cos(a)\, \cos(b) + \sin(a)\, \sin(b)\end{aligned}.

Simplify to obtain the formula:


\cos(a + b) + \cos(a - b) = 2\, \cos(a)\, \cos(b).

Before applying this sum-to-product formula to
\cos(x - 1) + \cos(x) = 0, it would be necessary to find the
a and
b such that:


  • a + b = x - 1.

  • a - b = x.

Solve this system of equations for
a and
b to obtain
a = x - (1/2) and
b = -(1/2).

Apply the sum-to-product formula:


\begin{aligned}& 2\, \cos(x - (1/2))\, \cos(-(1/2)) \\ =\; & 2\, \cos(a)\, \cos(b) \\ =\; & \cos(a + b) + \cos(a - b) && (\text{sum-to-product}) \\ =\; & \cos(x - (1/2) + (-1/2)) \\ &+ \cos(x - (1/2) - (-1/2)) \\ =\; & \cos(x - 1) + \cos(x) \\ =\; & 0\end{aligned}.

In other words, the original equation
\cos(x - 1) + \cos(x) = 0 is equivalent to
2\, \cos(x - (1/2))\, \cos(-(1/2)) = 0. Solve this new equation for
x:


2\, \cos(x - (1/2))\, \cos(-(1/2)) = 0.


\cos(x - (1/2)) = 0.


\text{$x - (1/2) = (k + (1/2))\, \pi$ for $k \in \mathbb{Z}$}.


\text{$x = (k + (1/2))\, \pi + (1/2)$ for $k \in \mathbb{Z}$}.

Therefore, the general solution to
\cos(x - 1) = -\cos(x) is
x = (k + (1/2))\, \pi + (1/2) for all
k \in \mathbb{Z}.

User Fred Pym
by
8.7k points
7 votes


~~~~\cos x-1= - \cos x\\\\\implies \cos x +\cos x = 1\\\\\implies 2 \cos x = 1 \\\\\implies \cos x =\frac 12\\\\\implies x = 2n\pi \pm\frac { \pi}3

User Nerkyator
by
8.7k points

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