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Wednesday, August 10, 2011

Math–Trigonometry Problem with Brilliant Solution!

Hey, I have found some good trigonometry problem and wanted to share with you my solution:
Find the x’s that satisfies the equation:
sqrt(2)*sin(x) – cos(x/2) = sin(x/2)
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 !

1 comment:

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



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