Question

In a ΔABC, right angled at A, if tan C = `sqrt3` , find the value of sin B cos C + cos B sin C.

Answer 1

In a Δle ABC right angled at A tan C = `sqrt3`

Find sin B cos C + cos B sin C

`tan c = sqrt3`

`tan c = "oppoite side"/"adjacent side"`

Let x be the hypotenuse By applying Pythagoras we get

`BC^2 = BA^2 + AC^2`

`x^2 = (sqrt3)^2 + 1^2`

x² = Δ ⇒ x = 2

At ∠B, `sin B = (AC)/(BC) = 1/2`

`cos B = sqrt3/2`

At ∠C, `sin = sqrt3/2`

`cos c = 1/2`

On substitution we get

`=> 1/2 xx 1/2 + sqrt3/2 xx sqrt3/2`

`=> 1/4 + (sqrt3)/4 xx (sqrt3) = (sqrt3 xx sqrt3 + 1)/4 = (3 + 1)/4 = 4/4 = 1`