How to solve cosx-2sinx=1cosx−2sinx=1 for 0<=x<=2pi0≤x≤2π?

2 Answers
Somebody N.
Feb 16, 2018

color(blue)(0,pi,2pi,5.356,4.070)color(white)(88)0,π,2π,5.356,4.07088 ( 3 .d.p.)

Explanation:

cosx-2sinx=1cosx−2sinx=1

Add 2sinx2sinx to both sides:

cosx=1+2sinxcosx=1+2sinx

Square both sides:

cos^2x=(1+2sinx)^2cos2x=(1+2sinx)2

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

cos^2x-(1+2sinx)^2=0cos2x−(1+2sinx)2=0

Using identity:

color(red)(bbsin^2x+cos^2x=1)sin2x+cos2x=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.)

Nghi N
Feb 16, 2018

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

Related questions
Impact of this question
9758 views around the world
You can reuse this answer
Creative Commons License