# Without using the calculator, find the exact value of tan(5pi/12).?

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My answer was tan 30 + tan 45

[ 3 + (square root of) 3 ] / 3.

is that right?

• tan( 5π/12) =

tan(3π/12 + 2π/12) =

tan( π/4 + π/6) =

[ tan(π/4) + tan(π/6) ] / [ 1 – tan(π/4)•tan(π/6)] =

[ 1 + 1/√3 ] / [ 1 – 1 • 1/√3] =

[ (3 + √3) /3 ] / [ (3 – √3) / 3] =

(3 + √3) / (3 – √3) =

(3 + √3)² / (9 – 3) =

(9 + 6√3 + 3) / 6 =

(12 + 6√3) / 6 =

2 + √3

see the link for a table of trig identities, in particular tangent of a sum.

• No you can’t just add.

Even though you can’t use calculator for finding solution, you should at least use calculator for verifying answer.

5π/12 = 75°

Actual value: tan (75°) = 3.732 (using calculator)

So this is obviously not right

Use angle sum identity:

tan(75°) = tan(45°+30°)

tan(75°) = (tan(45°)+tan(30°)) / (1 – tan(45°)*tan(30°))

tan(75°) = (1 + √3/3) / (1 – 1*√3/3)

tan(75°) = ((3+√3)/3) / ((3-√3)/3)

tan(75°) = (3+√3) / (3-√3)

tan(75°) = (3+√3)(3+√3) / (3-√3)(3+√3)

tan(75°) = (9+6√3+3) / (9-3)

tan(75°) = (12+6√3) / 6

tan(75°) = 2 + √3

Check using calculator

tan(75°) = 3.7320508075688772935274463415059

2 + √3 = 3.7320508075688772935274463415059

ok

• 30 + tan 45

[ 3 + (square root of) 3 ] / 3.

was the answer i got as well i am sure it is right

• no

ans = tan (30 +45) does not = tan 30 + tan 45

need formula that reduces tan(a+b) to simpler form

in terms of sin/ cos a and b

• Find Exact Value Of Tan

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