write the expression 1-cos 60/1+tan 30 in surd form and rationalize
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.
To simplify the expression 1 - cos 60 / 1 + tan 30 and write it in surd form, we can start by evaluating the trigonometric functions of 60 and 30 degrees.
1. cos 60 = 1/2
This value can be obtained from the unit circle or a trigonometric table.
2. tan 30 = √3 / 3
Tan can be calculated by taking the ratio of the sine and cosine functions: sin 30 / cos 30 = (1/2) / (√3/2) = √3 / 3.
Now let's substitute these values back into the expression:
1 - 1/2 / 1 + √3 / 3
Next, we need to simplify the expression, rationalize the denominator, and rewrite it in surd form:
To rationalize the denominator, we multiply the numerator and denominator of the fraction by the conjugate of the denominator (√3 - 3):
[(1 - 1/2) * (3 - √3)] / [(1 + √3 / 3) * (3 - √3)]
Simplifying the numerator:
[(2/2 - 1/2) * (3 - √3)] / [(1 + √3 / 3) * (3 - √3)]
[(1/2) * (3 - √3)] / [(1 + √3 / 3) * (3 - √3)]
(3/2 - √3/2) / (3 + √3)
Finally, we can multiply the numerator and denominator by the conjugate of the denominator (3 - √3) to eliminate the square root in the denominator:
[(3/2 - √3/2) * (3 - √3)] / [(3 + √3) * (3 - √3)]
[(9/2 - 3√3/2 - 3√3 + √3 * √3)] / (9 - 3)
[(9 - 6√3)] / 6
Therefore, the expression 1 - cos 60 / 1 + tan 30 in surd form and rationalized is:
(9 - 6√3) / 6