Question

How to solve `cosx-2sinx=1` for `0<=x<=2pi`?

Answer 1

`color(blue)(0,pi,2pi,5.356,4.070)color(white)(88)` ( 3 .d.p.)

Explanation:

`cosx-2sinx=1`

Add `2sinx` to both sides:

`cosx=1+2sinx`

Square both sides:

`cos^2x=(1+2sinx)^2`

Subract `(1+2sinx)^2` from both sides:

`cos^2x-(1+2sinx)^2=0`

Using identity:

`color(red)(bbsin^2x+cos^2x=1)`

`color(red)bb(cos^2x=1-sin^2x)`

`1-sin^2x-(1+2sinx)^2=0`

Expand bracket:

`1-sin^2x-(1+4sinx+4sin^2x)=0`

`1-sin^2x-1-4sinx-4sin^2x=0`

`-5sin^2x-4sinx=0`

`5sin^2x+4sinx=0`

Let `u=sinx`

Then:

`5u^2+4u=0`

Factor:

`u(5u+4)=0=>u=0 and u=-4/5`

But `u=sinx`

`sinx=0`

`sinx=-4/5`

`x=arcsin(sinx)=arcsin(0)=>x=0,pi,2pi`

`x=arcsin(sinx)=arcsin(-4/5)=>x=arcsin(-4/5)+2pi`

`x= pi-arcsin(-4/5)`

`color(blue)(0,pi,2pi,5.356,4.070)color(white)(88)` ( 3 .d.p.)

Answer 2

`x = 0^@`, and `x = 233.1^@`

Explanation:

cos x - 2sin x = 1
Call `tan a = sin a/(cos a) = 2` --> `a = 63^@43`
After cross multiplication -->
`cos x.cos a - sin a.sin x = cos a = cos 63^@43 = 0.45`
Using sum identity, we get:
`cos (x + a) = 0.45`
There are 2 solutions:
`x + a = +- 63^@43`
a. `x + 63.43 = 63^@43` --> `x = 0^@`
b. `x + 63.43 = - 63^@43` --> `x = - 126.87`, or
`x = 360 - 126.9 = 233^@13`
Check.
x = 0 --> cos x = 1 --> -2sin x = 0 -->
cos x - 2sin x = 1 . proved
x = 233.13 --> cos x = - 0.60 --> 2sin x = (2)(-0.8) = - 1.60 -->
--> cos x - 2sin x = - 0.6 + 1.6 = 1. Proved