__Find the x’s that satisfies the equation__:

sqrt(2)*sin(x) – cos(x/2) = sin(x/2)

__Solution__:

Let’s multiply by 1 the whole equation and it will be solved!

1 = tan(45 degrees) = tan(pi/4).

**Note for next calculations tan(45 degrees) will be just tan(45)**.

__So we get__:

sin(x/2) + cos(x/2) = 2*sqrt(2)*sin(x/2)*cos(x/2) #

**# We just use the simple trigonometric identity: sin(2*x) = 2*sin(x)*cos(x).**

__Now using the Brilliant trick__:

tan(45)*sin(x/2) + 1*cos(x/2) = 2*sqrt(2)*1*sin(x/2)*cos(x/2) #

**#**

**We use tan(45) sometimes like 1 and sometimes like tan(45) as is.**

__Now just some little technique but the exercise is done__:

Note: tan(45) = sin(45)/cos(45)

Using the note, applying it on what we have done already:

[sin(45)/cos(45)] * sin(x/2) + cos(x/2) = 2*sqrt(2)*sin(x/2)*cos(x/2)

__Multiply by cos(45)__:

sin(45)*sin(x/2) + cos(45)*cos(x/2) = 2*sin(x/2)*cos(x/2) #

**#**

**cos(45) = sqrt(2)/2 so when you multiply it by sqrt(2) you get 1.**

__Now use the known identity of adding angles__:

sin(45)*sin(x/2) + cos(45)*cos(x/2) = sin(x) [Remember: sin(2*x) = 2*sin(x)*cos(x)]

Now using: cos(x-y) = cos(x)*cos(y) + sin(x)*sin(y)

**#**

**Other Identities.**

__Now the final step__:

__Using the identity above__:

cos(x/2 – 45) = sin(x)

__And the x’s are__:

x/2 – 45 = 90 – x + 360*k where k is integer.

__And you will get that__:

x = 90 +240*k or x = 90 + 360k where k is again integer.

VoilĂ !

You may have made this a bit more complicated than necessary. Just dividing the original equation by sqrt(2) and collecting terms gives

ReplyDeletesin(x)=(1/sqrt(2))sin(x/2)+(1/sqrt(2))cos(x/2)

so sin(x)=sin(x/2 + pi/4)

so either x/2+pi/4=x +k*2pi

(giving x=pi/2 +4kpi or in degrees 90+720k)

or x/2+pi/4=pi-x +k*2pi

(giving x=pi/2 +4kpi/3 or in degrees 90+240k)