Solving trig equation: cos2x cosx - sin2x sinx = -1
- Thread starter d4dewu
- Start date
It requires the use of some identities.
\(\displaystyle \L\\cos(x)cos(2x)-sin(x)sin(2x)=-1\)
identities: \(\displaystyle cos(2x)=2cos^{2}(x)-1 \;\ and \;\ sin(2x)=2sin(x)cos(x)\)
=\(\displaystyle \L\\2cos^{3}(x)-cos(x)-2sin^{2}(x)cos(x)=-1\)
=\(\displaystyle \L\\cos(x)(2cos^{2}(x)-1-2sin^{2}(x))=-1\)
identity: \(\displaystyle sin^{2}(x)=cos^{2}(x)-1\)
=\(\displaystyle \L\\cos(x)(4cos^{2}(x)-3)=-1\)
=\(\displaystyle \L\\4cos^{3}(x)-3cos(x)=-1\)
Let u=cos(x)
Then you have:
\(\displaystyle \L\\4u^{3}-3u+1=0\)
Factor:
\(\displaystyle \L\\(u+1)(2u-1)^{2}=0\)
\(\displaystyle u=-1 \;\ and \;\ u=\frac{1}{2}\)
\(\displaystyle \L\\cos^{-1}(-1)={\pi}=180\)
\(\displaystyle \L\\cos^{-1}(\frac{1}{2})=\frac{\pi}{3}=60\)
\(\displaystyle \L\\cos^{-1}(\frac{1}{2})=\frac{5{\pi}}{3}=300\)
\(\displaystyle \L\\cos(x)cos(2x)-sin(x)sin(2x)=-1\)
identities: \(\displaystyle cos(2x)=2cos^{2}(x)-1 \;\ and \;\ sin(2x)=2sin(x)cos(x)\)
=\(\displaystyle \L\\2cos^{3}(x)-cos(x)-2sin^{2}(x)cos(x)=-1\)
=\(\displaystyle \L\\cos(x)(2cos^{2}(x)-1-2sin^{2}(x))=-1\)
identity: \(\displaystyle sin^{2}(x)=cos^{2}(x)-1\)
=\(\displaystyle \L\\cos(x)(4cos^{2}(x)-3)=-1\)
=\(\displaystyle \L\\4cos^{3}(x)-3cos(x)=-1\)
Let u=cos(x)
Then you have:
\(\displaystyle \L\\4u^{3}-3u+1=0\)
Factor:
\(\displaystyle \L\\(u+1)(2u-1)^{2}=0\)
\(\displaystyle u=-1 \;\ and \;\ u=\frac{1}{2}\)
\(\displaystyle \L\\cos^{-1}(-1)={\pi}=180\)
\(\displaystyle \L\\cos^{-1}(\frac{1}{2})=\frac{\pi}{3}=60\)
\(\displaystyle \L\\cos^{-1}(\frac{1}{2})=\frac{5{\pi}}{3}=300\)
skeeter
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look at the pattern of the left side of the equation ...
cos(2x)cos(x) - sin(2x)sin(x)
... look familiar? it follows the pattern for the cosine sum identity
cos(a)cos(b) - sin(a)sin(b) = cos(a + b)
cos(2x)cos(x) - sin(2x)sin(x) = cos(2x + x) = cos(3x) = -1
since 0 < x < 360, 0 < 3x < 1080
between 0 and 1080 degrees, cosine has a value of -1 at 180, 540, and 900 degrees
3x = 180, 540, 900
x = 60, 180, 300
cos(2x)cos(x) - sin(2x)sin(x)
... look familiar? it follows the pattern for the cosine sum identity
cos(a)cos(b) - sin(a)sin(b) = cos(a + b)
cos(2x)cos(x) - sin(2x)sin(x) = cos(2x + x) = cos(3x) = -1
since 0 < x < 360, 0 < 3x < 1080
between 0 and 1080 degrees, cosine has a value of -1 at 180, 540, and 900 degrees
3x = 180, 540, 900
x = 60, 180, 300