The value of cos([tex]60^{o}[/tex]) = [tex]\frac{1}{2}[/tex]
We have : sin [tex]30^{o}[/tex] = [tex]\frac{1}{2}[/tex]
We have to find the value cos [tex]60^{o}[/tex] using the value of sin ([tex]30^{o}[/tex]).
We can express cos(2x) in terms of cos(x) and sin(x) as follows -
[tex]cos(2x)=cos^{2}x - sin^{2}x\\\\[/tex]. (Eqn. 1)
We know that -
[tex]1 = cos^{2}x +sin^{2}x[/tex] (Eqn. 2)
Add both equations 1 and 2, we get -
[tex]2\;cos^{2}x =1+cos 2(x)[/tex]
[tex]cos\; x = \sqrt{\frac{1+cos(2x) }{2} }[/tex]
We can write cos ([tex]60^{o}[/tex]) as cos (2 x [tex]30^{o}[/tex]). Hence , x = [tex]30^{o}[/tex]
cos (2 x [tex]30^{o}[/tex]) = [tex]cos^{2}\;30^{o} - sin^{2}\;30^{o}[/tex] = [tex](\frac{\sqrt{3} }{2}) ^{2} - (\frac{1}{2}) ^{2}[/tex] = [tex]\\\\\frac{3}{4} - \frac{1}{4}[/tex] = [tex]\frac{1}{2}[/tex]
Hence, the value of cos([tex]60^{o}[/tex]) = [tex]\frac{1}{2}[/tex]
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