express the square root of 3 divided by 1-cos30 degrees in surd form and simplify without using a calculator or mathematical tables
To express the square root of 3 divided by 1 - cos(30 degrees) in surd form and simplify it, we need to make use of the trigonometric identity: 1 - cos(x) = 2sin^2(x/2).
In this case, x = 30 degrees, so we have:
1 - cos(30 degrees) = 2sin^2(30 degrees / 2)
Now, we can substitute this into the expression to get:
√3 / (1 - cos(30 degrees)) = √3 / (2sin^2(30 degrees / 2))
Since sin(30 degrees) = 1/2, we can simplify further:
√3 / (1 - cos(30 degrees)) = √3 / (2(1/2))^2
= √3 / (2(1/4))
= √3 / (1/2)
= √3 * (2/1)
= 2√3
Therefore, the square root of 3 divided by 1 - cos(30 degrees) simplified in surd form is 2√3.
To simplify the expression √3 / (1 - cos30°) in surd form, we need to work on simplifying the denominator first.
The first step is to find the value of cos30°. We know that cos30° = √3/2.
Now, let's substitute this value into the denominator:
1 - cos30° = 1 - √3/2
To simplify the denominator, we need to rationalize it. To do this, we can multiply the numerator and denominator by the conjugate of the denominator, which is 2:
(1 - √3/2) x (2/2) = (2 - √3)/2
Now we have:
√3 / (1 - cos30°) = √3 / ((2 - √3)/2)
To divide by a fraction, we invert the denominator and multiply:
√3 / ((2 - √3)/2) = √3 x (2/ (2 - √3))
To simplify further, we can multiply the numerator and denominator by the conjugate of the denominator, which is (2 + √3):
√3 x (2/ (2 - √3)) x ((2 + √3)/ (2 + √3))
Expanding this expression, we get:
(2√3 + 3)/ (4 - 3)
Simplifying the denominator:
(2√3 + 3)/ 1
So, the simplified surd form of √3 / (1 - cos30°) is 2√3 + 3.