Question

Find the value of:

`(4 sin 60° + (cos 60°)/(sin 30°)) (4 cos 30° − 2 sin 30°)` 

Answer 1

Given:

`(4 sin 60° + (cos 60°)/(sin 30°)) (4 cos 30° − 2 sin 30°)` 

Step 1: Use standard trigonometric values

sin ⁡60° = `sqrt3/2`

cos ⁡60° = `1/2`

sin⁡ 30° = `1/2`

cos⁡ 30° = `sqrt3/2`

Step 2: Substitute the values

4 · `sqrt3/2 + (1/2)/(1/2) = 2 sqrt3 + 1`

4 · `sqrt3/2 − 2 · 1/2 = 2 sqrt3 − 1`

Step 3: Multiply the expressions

`(2 sqrt3 + 1) (2 sqrt3 − 1)`

(a + b) (a − b) = a² − b²   ...[converted into this form]

`(2 sqrt3)^2 − (1)^2 = 4 xx 3` − 1 = 12 − 1

 = 11