How do you solve $sin (x + (π/4)) + sin (x - (π/4)) = 1$ ?

Question

6621 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-07-22T12:11:02

$x=(-1)^n(pi/4)+npi" ", n in ZZ$

We use the identity(otherwise called the Factor Formula) :

$sinA +sinB=2sin((A+B)/2)cos((A-B)/2)$

Like this :

$sin (x + (pi/4)) + sin (x - (pi/4)) = 2sin[((x+pi/4)+(x-pi/4))/2]cos[(x+pi/4-+(x-pi/4))/2]= 1$

$=>2sin((2x)/2)cos((2*(pi/4))/2)=1$

$=>2sin(x)cos(pi/4)=1$

$=>2*sin(x)*sqrt(2)/2=1$

$=>sin(x)=1/sqrt(2)=sqrt(2)/2$

$=>color(blue)(x=pi/4)$

The General Solution is : $x= pi/4 + 2pik$ and $x=pi-pi/4 +2pik=pi/4 + (2k+1)pi" ",k in ZZ$

You can combine the two sets of solution into one as follows :

$color(blue)(x=(-1)^n(pi/4)+npi)" ",n in ZZ$

How do you solve sin (x + (π/4)) + sin (x - (π/4)) = 1sin (x + (π/4)) + sin (x - (π/4)) = 1?