Question

Evaluate: `(sin^3 60^circ - tan 30^circ)/(cos^2 45^circ)`

Answer 1

1. Identify trigonometric values

First, recall the standard values for the trigonometric ratios at the given angles:

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

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

`sin 45^circ = 1/sqrt(2)`

2. Substitute and calculate terms

Substitute these values into each part of the expression:

`sin^3 60^circ: (sqrt(3)/2)^3 = (3sqrt(3))/sqrt(8)`

`cos^2 45^circ: (1/sqrt(2))^2 = 1/2`

3. Simplify the numerator

Subtract tan 30° from sin³ 60°:

`(3sqrt(3))/8 - 1/sqrt(3)`

To subtract, find a common denominator (which is `8sqrt(3)`):

`(3sqrt(3) xx sqrt(3) - 1 xx 8)/(8sqrt(3))`

= `(3(3) - 8)/(8sqrt(3))`

= `(9 - 8)/(8sqrt(3))`

= `1/(8sqrt(3))`

4. Divide by the denominator

Divide the simplified numerator by the simplified denominator:

`(1/(8sqrt(3)))/(1/2)`

= `1/(8sqrt(3)) xx 2`

= `1/(4sqrt(3))`

5. Rationalize the result

Multiply the numerator and denominator by `sqrt(3)` to remove the radical from the bottom:

`1/(4sqrt(3)) xx sqrt(3)/sqrt(3)`

= `sqrt(3)/(4 xx 3)`

= `sqrt(3)/12`