Question

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

Answer 1

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