Advanced Trigonometry help needed, sinx = -4cosx... anyone?

sinx = -4cosx

divide by cosx

sinx/cosx = -4

tanx = -4

x = 104 or 284 degrees in the interval 0-360 degrees

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Advanced Trigonometry help needed, sinx = -4cosx... anyone?

I am going through a past Additional Maths (OCR) paper, and I have no idea how to do this question;

Find all the values of x in the range of 0 < x < 360 (degrees) that satisfy sin x = -4 cos x.

The question is worth 5 marks on the paper!

Additionally, any good revision sites for...

Let x=arccosy.

By sin(arccosθ)=√(1-θ^2), we get sinx=sin(arccosy)=√(1-y^2).

So we can derive by arccos(cosθ)=θ, -4cosx=-4y.

√(1-y^2)=-4y

Squaring both sides gives

1-y^2=16y^2

Adding y^2 to both sides gives

1=17y^2

So y = 1/√17

So x=arccos(1/√17).

I'd say, therefore, 1 value. I'm checking it though, so I might turn out some mistake.

----------edit--------------------

Mαtt is actually correct, but I don't know where his argument or mine fell apart. His answer relies on the fact that tanθ=sinθ/cosθ.

sinx = -4 cosx

Dividing both sides by cosx yields

sinx/cosx=-4

Applying tanθ=sinθ/cosθ,

tanx=-4

Taking the arctans of both sides to invert tan gives

x=arctan(-4).

I think that the fallacy occurs in that I create x off another variable, but I still can't spot any errors.

sinx = -4cosx

=>sinx+4cosx=0

=>(sinx+4cosx)/√[1²+4²]=0

=>(sinx+4cosx)/√17=0

{

let cosΘ=1/√17

sin²Θ=1-cos²Θ

=1-1/17=16/17

=>sinΘ=4/√17

}

=>sinxcosΘ+cosxsinΘ=0

=>sin(x+Θ)=0

=>x+Θ=0,180,360

=>x=-Θ or 180-Θ or 360-Θ

only the last two lie in the intervel

thus

x=180-Θ or 360-Θ whr Θ=cos-1(1/√17)=75.963756532073521417107679840844

x=180-76=104

or

x=360-76=284

Or simply

sin x = -4 cos x.

tanx=-4

x=tan-1(-4)

-75.963756532073521417107679840837

180+of that

or 360+of that

so we get the same ans

x=104 or 284

sin x+4 cos x=0

exist a number a with propriety :

cos(a) =1/sqrt(17)

and

sin(a)=4/sqrt(17)

a exist because

(1/sqrt(17))^2+(4/sqrt(17))^2=1

then I rewrite :

sin x* cos(a)+sin a*cos x=0

sin(x+a)=0

from here :

x+a=k pi (k any integer, pi=3.1415......)

x=-a+kpi

and a = arcsin(4/sqrt(17))

sin(x) = -4cos(x)

tan(x) = -4

x = arctan(-4)

Which does not have a solution on the range 0 < x < 360

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