Question
write the expression 1-cos 60/1+tan 30 in surd form and rationalize
Answers
GPT 3.5
To write the expression \(1-\frac{\cos 60}{1+\tan 30}\) in surd form, we need to simplify it.
First, let's evaluate \(\cos 60\) and \(\tan 30\).
We know that \(\cos 60 = \frac{1}{2}\) and \(\tan 30 = \frac{\sin 30}{\cos 30} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\).
Substituting these values in the expression, we get:
\(1 - \frac{1/2}{1 + \frac{1}{\sqrt{3}}}\)
Now, to rationalize the denominator, we can multiply both the numerator and denominator by the conjugate of the denominator, which is \(1 - \frac{1}{\sqrt{3}}\) (the negative sign is used because we want to change the sign in the denominator).
This gives us:
\[\frac{\left(1 - \frac{1/2}{1 + \frac{1}{\sqrt{3}}}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\left(1 + \frac{1}{\sqrt{3}}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}\]
Simplifying further, we have:
\[\frac{\left(\frac{2}{2} - \frac{1}{2(1 + \frac{1}{\sqrt{3}})}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\left(\frac{\sqrt{3}}{\sqrt{3}} + \frac{1}{\sqrt{3}}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}\]
\[\frac{\left(\frac{1}{2(1 + \frac{1}{\sqrt{3}})}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\left(\frac{\sqrt{3} + 1}{\sqrt{3}}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}\]
\[\frac{\left(\frac{1}{1 + \frac{1}{\sqrt{3}}} - \frac{1}{2(1 + \frac{1}{\sqrt{3}})}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\left(\frac{(\sqrt{3} + 1)(\sqrt{3})}{\sqrt{3}}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}\]
\[\frac{\left(\frac{\sqrt{3}}{\sqrt{3} + 1} - \frac{\sqrt{3}}{2(\sqrt{3} + 1)}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\left((\sqrt{3} + 1)\left(\frac{2\sqrt{3}\ - 1}{\sqrt{3}}\right)\right) \left(1 - \frac{1}{\sqrt{3}}\right)}\]
\[\frac{\left(\frac{\sqrt{3}}{\sqrt{3} + 1} - \frac{\sqrt{3}}{2\sqrt{3} + 2}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\left(\frac{(2\sqrt{3}\ - 1)(\sqrt{3} + 1)}{\sqrt{3}}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}\]
\[\frac{\left(\frac{\sqrt{3}}{\sqrt{3} + 1} - \frac{\sqrt{3}}{\sqrt{3} + 2}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\frac{(2\sqrt{3}\ - 1)(\sqrt{3} + 1)}{\sqrt{3}} \cdot \frac{\sqrt{3} - 1}{\sqrt{3}}}\]
\[\frac{\left(\frac{\sqrt{3}}{\sqrt{3} + 1} - \frac{\sqrt{3}}{\sqrt{3} + 2}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\frac{(2\sqrt{3}\ - 1)(\sqrt{3} + 1)(\sqrt{3} - 1)}{3}}\]
Further simplifications can be done by expanding and combining like terms in the numerator and denominator, but this is the final expression in surd form.
First, let's evaluate \(\cos 60\) and \(\tan 30\).
We know that \(\cos 60 = \frac{1}{2}\) and \(\tan 30 = \frac{\sin 30}{\cos 30} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\).
Substituting these values in the expression, we get:
\(1 - \frac{1/2}{1 + \frac{1}{\sqrt{3}}}\)
Now, to rationalize the denominator, we can multiply both the numerator and denominator by the conjugate of the denominator, which is \(1 - \frac{1}{\sqrt{3}}\) (the negative sign is used because we want to change the sign in the denominator).
This gives us:
\[\frac{\left(1 - \frac{1/2}{1 + \frac{1}{\sqrt{3}}}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\left(1 + \frac{1}{\sqrt{3}}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}\]
Simplifying further, we have:
\[\frac{\left(\frac{2}{2} - \frac{1}{2(1 + \frac{1}{\sqrt{3}})}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\left(\frac{\sqrt{3}}{\sqrt{3}} + \frac{1}{\sqrt{3}}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}\]
\[\frac{\left(\frac{1}{2(1 + \frac{1}{\sqrt{3}})}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\left(\frac{\sqrt{3} + 1}{\sqrt{3}}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}\]
\[\frac{\left(\frac{1}{1 + \frac{1}{\sqrt{3}}} - \frac{1}{2(1 + \frac{1}{\sqrt{3}})}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\left(\frac{(\sqrt{3} + 1)(\sqrt{3})}{\sqrt{3}}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}\]
\[\frac{\left(\frac{\sqrt{3}}{\sqrt{3} + 1} - \frac{\sqrt{3}}{2(\sqrt{3} + 1)}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\left((\sqrt{3} + 1)\left(\frac{2\sqrt{3}\ - 1}{\sqrt{3}}\right)\right) \left(1 - \frac{1}{\sqrt{3}}\right)}\]
\[\frac{\left(\frac{\sqrt{3}}{\sqrt{3} + 1} - \frac{\sqrt{3}}{2\sqrt{3} + 2}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\left(\frac{(2\sqrt{3}\ - 1)(\sqrt{3} + 1)}{\sqrt{3}}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}\]
\[\frac{\left(\frac{\sqrt{3}}{\sqrt{3} + 1} - \frac{\sqrt{3}}{\sqrt{3} + 2}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\frac{(2\sqrt{3}\ - 1)(\sqrt{3} + 1)}{\sqrt{3}} \cdot \frac{\sqrt{3} - 1}{\sqrt{3}}}\]
\[\frac{\left(\frac{\sqrt{3}}{\sqrt{3} + 1} - \frac{\sqrt{3}}{\sqrt{3} + 2}\right) \left(1 - \frac{1}{\sqrt{3}}\right)}{\frac{(2\sqrt{3}\ - 1)(\sqrt{3} + 1)(\sqrt{3} - 1)}{3}}\]
Further simplifications can be done by expanding and combining like terms in the numerator and denominator, but this is the final expression in surd form.