How do you solve for x in cos(-100)= cos(55)cos x+sin(55)sin x?

Jul 21, 2018

$x \in \left\{360 k + 155\right\} \cup \left\{360 k - 45\right\} , k \in \mathbb{Z}$.

Explanation:

Given eqn. is, $\cos 55 \cos x + \sin 55 \sin x = \cos \left(- 100\right)$.

$\therefore \cos \left(x - 55\right) = \cos 100$.

But, $\cos \theta = \cos \alpha \Rightarrow \theta = 2 k \pi \pm \alpha , k \in \mathbb{Z}$.

$\therefore \cos \left(x - 55\right) = \cos 100 \Rightarrow x - 55 = 360 k \pm 100 , k \in \mathbb{Z}$.

$\therefore x = 360 k + 55 \pm 100 , k \in \mathbb{Z}$.

$\therefore x \in \left\{360 k + 155\right\} \cup \left\{360 k - 45\right\} , k \in \mathbb{Z}$.

$x = 55 + 2 n \setminus \pi \setminus \pm 100$

Where, $n$ is any integer

Explanation:

Given trigonometric equation:

$\setminus \cos \left(- 100\right) = \setminus \cos \left(55\right) \setminus \cos x + \setminus \sin \left(55\right) \setminus \sin x$

$\setminus \cos \left(100\right) = \setminus \cos \left(55 - x\right)$

$\setminus \cos \left(55 - x\right) = \setminus \cos \left(100\right)$

$\setminus \cos \left(x - 55\right) = \setminus \cos \left(100\right)$

$x - 55 = 2 n \setminus \pi \setminus \pm 100$

$x = 55 + 2 n \setminus \pi \setminus \pm 100$

Where, $n$ is any integer i.e. $n = 0 , \setminus \pm 1 , \setminus \pm 2 , \setminus \pm 3 , \setminus \ldots$