write the expression 1-cos 60/1+tan 30 in surd form and rationalize the denominator

Answer 1

To simplify the given expression and rationalize the denominator, we'll start by simplifying the individual trigonometric functions:

1. Simplify cosine of 60 degrees:
Recall that cos(60) = 1/2.
So, cos 60 = 1/2.

2. Simplify tangent of 30 degrees:
Recall that tan(30) = sin(30)/cos(30).
Since sin(30) = 1/2 and cos(30) = √3/2, we have tan(30) = (1/2) / (√3/2) = 1/√3 = √3/3.

Now, substitute the simplified trigonometric values back into the expression:

1 - cos(60) / (1 + tan(30))
= 1 - (1/2) / (1 + √3/3)

To rationalize the denominator, we'll multiply the numerator and denominator by the conjugate of the denominator:
= (1 - (1/2)) * (3/3) / ((1 + √3/3) * (3/3))
= (2/2 - 1/2) * 3 / (3 + √3)

Simplifying further:
= (1/2) * 3 / (3 + √3)
= 3/2 / (3 + √3)

Thus, the expression 1 - cos(60) / (1 + tan(30)) expressed in surd form and with the denominator rationalized is 3/ (2(3 + √3)).