write the expression 1-cos 60/1+tan 30 in surd form and rationalize the denominator
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)).
To write the expression 1 - cos 60/(1 + tan 30) in surd form, we need to simplify the fractions and eliminate any trigonometric functions.
First, let's evaluate the trigonometric functions:
cos 60 = 1/2
tan 30 = sin 30 / cos 30
Since sin 30 = 1/2 and cos 30 = √3/2, tan 30 = (1/2) / (√3/2) = 1/√3.
Substituting these values back into the expression:
1 - cos 60 / (1 + tan 30) = 1 - (1/2) / (1 + 1/√3)
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of (1 + 1/√3), which is (1 - 1/√3):
(1 - (1/2)) * (1 - 1/√3) / (1 + 1/√3) * (1 - 1/√3)
Expanding and simplifying the expression:
(1/2) * (√3 - 1) / (1 - 1/3) = (1/2) * (√3 - 1) / (2/3) = 3/4 * (√3 - 1)
Therefore, the expression 1 - cos 60 / (1 + tan 30) in surd form is 3/4 * (√3 - 1).