Finding solutions of an equation

Sep 2017
1
0
Manchester
Find all of the solutions of the equation

cos(56°-3x)=-0.785

where 0
° ≤ x ≤ 360°


I've been able to find one soloution using
x = (56 - arccos(-0.785)) / 3. But I'm unable to find any others.

Thanks in advance for any help
 
Aug 2017
34
10
Finland
Since cos is oscillating function, $\cos(x)=a$ has infinite solutions.
For any integer $n$, due to the fact that $\cos(2n\pi\pm\theta)=\cos(\theta)$, all the solutions of above equation can be written by, $x = 2n\pi \pm \arccos(a)$.
Therefore, your problem of $\cos(\theta-kx)=a=\cos(kx-\theta)$ has solutions in the form of $x = \frac{2n\pi \pm \arccos(a) + \theta}{k}$ for all integer $n$.
[Note that $\theta$ should be in Radians, not degrees]
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
Do you know what the graph of y= cos(x) looks like? cos(360- x)= cos(x). The principle solution to cos(x)= 0.785, in degrees, is 38.3 degrees. So 360- 38.3= 321.7 degrees is the other solution between 0 and 360 degrees. With 56- 3x= 321.7, x= (321.7- 56)/(-3)= -88.6 degrees.

(zemozamster, the original problem is given in degrees so the answer should be in degrees.)
 
Aug 2017
34
10
Finland
(zemozamster said:
That is why the answer is given in a generic form. Radians to degree conversion is something that he/she require to have before coming to this type of problems.
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
cos(56°-3x)=-0.785

where 0° ≤ x ≤ 360°


cosine is an even function $\implies \cos(56-3x) = \cos(3x-56)$

$0 \le x < 360 \implies 0 \le 3x < 1080 \implies -56 \le 3x-56 < 1024$

let $u=3x-56$. $\cos{u} = -0.785 \implies u$ is an angle residing in quadrants II or III

Quad II angles ...$u = \arccos(-0.785) \approx 141.72$ and the additional coterminal angles $501.72$ and $861.72$

$3x-56= \{141.72,501.72,861.72 \} \implies x \in \{65.9, 185.9, 305.9 \}$

Quad III angles ... $u = 360 - \arccos(-0.785) \approx 218.28$ and the additional coterminal angles $578.28$ and $938.28$

$3x-56= \{218.28,578.28,938.28 \} \implies x \in \{91.43, 211,43, 331.43 \}$
 

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