How do you find the exact functional value tan(45°+30°) using the cosine sum or difference identity?

1 Answer
Oct 16, 2015

Find exact value of tan (45 + 30).

Ans: #(sqrt3 + 1)/(sqrt3 - 1)#

Explanation:

#tan (45 + 30) = (sin (45 + 30))/(cos (45 + 30))#
Find sin (45 + 30) and cos (45 + 30)
sin (45 + 30) = sin 45.cos 30 + sin 30.cos 45 =
#= (sqrt3/2)(sqrt2/2) + (1/2)(sqrt2/2) = sqrt2/4(sqrt3 + 1)#

cos (45 + 30) = cos 45.cos 30 - sin 45.sin 30 =
#= (sqrt3/2)(sqrt2/2) - (1/2)(sqrt2/2) = sqrt2/4(sqrt3 - 1)#

#tan (45 + 30) = (sin)/(cos) = (sqrt3 + 1)/(sqrt3 - 1)#.

Check by calculator.
tan (45 + 30) = tan (75) = 3.73
#(sqrt3 + 1)/(sqrt3 - 1) = 2.73/0.73 = 3.73#. OK