I am unable to solve angle of a trigonometric equation

Question

I am fairly new to Mathematica, so this may be a simplistic question, but I want to learn. I know how to use the Solve function to solve a single equation like this:

$$\cos[41.614-x]-1.2752|cos[1.607x]+7.1644\sin[1.607x]=0$$

and I know how to use Solve to solve a trigonometric equation like this:

equation=1-2Cos[x]-2Sin[x]+4Sin[2x]==0
Solve[equation,x]

However, when I try to solve my equation using Solve or NSolve Nothing comes up. I use following code

equation=Cos[41.614−x]−1.2752*Cos[1.607x]+7.1644*sin[1.607x]==0
Solve[equation,x]

What do I need to do to get Mathematica to give me a value of theta in degrees? I am sure there is a straightforward solution, but I just can't seem to find it. Thank you!

Answer 1

If you fix sin to Sin and help NSolve (not Solve) a little by giving range of x then it can do it

ClearAll[x]; 
equation = Cos[41.614 - x] - 1.2752*Cos[1.607*x] + 7.1644*Sin[1.607*x] == 0; 
NSolve[equation && -4*Pi < x < 4*Pi, x]

gives

{{x -> -11.5371}, {x -> -9.61635}, {x -> -7.76423}, {x -> -5.83644}, \
{x -> -3.81118}, {x -> -1.77431}, {x -> 0.180699}, {x -> 
   2.03885}, {x -> 3.93373}, {x -> 5.93711}, {x -> 7.98094}, {x -> 
   9.96783}, {x -> 11.8458}}

Solve can't solve this type of equation. Need to use numerical solver.