Question

Evaluate: `(2 sin^2 60^circ + cos^2 60^circ)/(tan^2 30^circ)`

Answer 1

1. Identify standard values

First, recall the values of the trigonometric ratios for the given angles:

`sin 60^circ = sqrt(3)/2`

`cos 60^circ = 1/sqrt(2)`

`tan 30^circ = 1/sqrt(3)`

2. Substitute into the numerator

Evaluate the numerator 2sin² 60° + cos² 60°:

`2(sqrt(3)/2)^2 + (1/2)^2`

`2(3/4) + 1/4`

`6/4 + 1/4 = 7/4`

3. Substitute into the denominator

Evaluate the denominator tan² 30°:

`(1/sqrt(3))^2 = 1/3`

4. Divide the numerator by denominator

Combine the simplified parts:

`(7//4)/(1//3) = 7/4 xx 3`

= `21/4`

In decimal form, this is 5.25.