Respuesta :
Answer:
Exact value of Cos(45° - 60°) is 0.96 using difference of two angles.
Step-by-step explanation:
Given:
Cos(45° - 60°)
We have to apply the formula of cosine for difference of the two angles.
Formula:
[tex]cos(a-b)=cos(a)\ cos(b)+sin(a)\ sin(b)[/tex]
Plugging the values.
⇒ [tex]cos(45-60)=cos(45)\ cos(60) + sin(45)\ sin(60)[/tex]
We know that the values :
[tex]sin(45) =cos(45) = \frac{1}{\sqrt{2} }[/tex]
[tex]sin(60)=\frac{\sqrt{3} }{2}[/tex] and [tex]cos(60)=\frac{1}{2}[/tex]
So,
⇒ [tex]cos(45-60)=(\frac{1}{\sqrt{2} } \times \frac{1}{2} ) + (\frac{1}{\sqrt{2} } \times \frac{\sqrt{3} }{2})[/tex]
⇒ [tex]cos(45-60)=(\frac{1}{2\sqrt{2} } + \frac{\sqrt{3} }{2\sqrt{2} })[/tex]
⇒ [tex]cos(45-60)=(\frac{1+\sqrt{3} }{2\sqrt{2} } )[/tex]
⇒ [tex]cos(45-60)=(\frac{1+\sqrt{3} }{2\sqrt{2} } )\times \frac{2\sqrt{2} }{2\sqrt{2} }[/tex] ...rationalizing
⇒ [tex]cos(45-60)=\frac{2\sqrt{2} +2\sqrt{6} }{ 8}[/tex]
⇒ [tex]cos(45-60)=\frac{2(\sqrt{2}+\sqrt{6})}{8}[/tex] ...taking 2 as a common factor
⇒ [tex]cos(45-60)=\frac{(\sqrt{2}+\sqrt{6})}{4}[/tex]
To find the exact values we have to put the values of sq-rt .
As, [tex]\sqrt{2}=1.41[/tex] and [tex]\sqrt{6} =2.44[/tex]
Then
⇒ [tex]cos(45-60)=\frac{( 1.41+2.44)}{4}[/tex]
⇒ [tex]cos(45-60)=\frac{( 3.85)}{4}[/tex]
⇒ [tex]cos(45-60)=0.96[/tex]
So the exact value of Cos(45° - 60°) is 0.96 using difference of two angles.
