Identities?

Question

Find the exact value of tan 255°.

a. (3 + sqrt of 3)/(3 - sqrt of 3)
b.(1+ sqrt of 3)/(1 - sqrt of 3)
c. 3.73205
d. (3 - sqrt of 3)/(3 + sqrt of 3)

Answer 1

Well, it's (a), but a FAR SIMPLER way of writing it is:

2 + sqrt 3 (= 3.732050809...).

I got this by saying that tan 255° = tan 75° = 1/tan 15° = 1/t, say.

Then, by using the "tan twice angle formula", i.e. by writing that tan 30° = 2t / (1 - t^2) and solving the resulting quadratic equation for t, I obtained:

t = 2 - sqrt 3, so that 1/t = 1/(2 - sqrt 3).

Then, multiplying top and bottom by 2 + sqrt 3 to rationalize the denominator, one gets

1/t = 2 + sqrt 3.

Meanwhile, your expression for (a) is (3 + sqrt 3)/(3 - sqrt 3).

Multiplying top and botttom of THAT by (3 + sqrt 3), ITS denominator rationalizes, leaving

(3 + sqrt 3)^2 / 6 = (9 + 3 + 6 sqrt 3) / 6 = 2 + sqrt 3,

which is EXACTLY what I found for tan 255° = tan 75°
= 1/tan 15° = 1/t, above.

QED

Live long and prosper.

P.S. I suppose that there MUST be another way of attacking the problem that yields the solution in the peculiar form in which it is expressed in answer (a), but for the life of me I can't see why on Earth one would leave it in that extremely awkward form when the denominator can be so readily and easily rationalized!

Answer 2

tan 255° = tan 75° by periodicity.

From the double angle formula we know
cos 150° = 2 cos^2 75° - 1 = 1 - 2 sin^2 75°
and we know cos 150° = -cos 30° = -√3 / 2.
So 2 cos^2 75° = 1 - √3 / 2
and 2 sin^2 75° = 1 + √3 / 2
So tan^2 75° = (1 + √3 / 2) / (1 - √3 / 2)
= (1 + √3 / 2)^2 / (1 - 3/4)
= 4 (1 + √3 / 2)^2
So tan 75° = √(tan^2 75°) [since we know tan 75° is positive]
= 2 (1 + √3 / 2)
= 2 + √3.

Now note that (3 + √3) / (3 - √3)
= (3 + √3)^2 / (9 - 3)
= (9 + 6√3 + 3) / 6
= 2 + √3.

So answer A is correct.

Incidentally, simply knowing that the answer should be positive is enough to rule out (b), and knowing that it should be greater than 1 is enough to rule out (d), and the word "exact" is enough to rule out (c). ;-)

Answer 3

tan 255° = tan 75°
= tan (45 + 30)°
= (1 + 1/√3) / (1 - 1 / √3)
= (√3 + 1) / (√3 - 1)
None of given options?

Answer 4

tan 255° = tan(180°+75°) = tan (75°)

=tan(30°+45°)

= [tan(30°) +tan(45°) ]/[ 1 - tan(30°) tan(45°)]

=[(1/√3) + 1] / [1 - (1/√3) ]

=[1 +√3 ] / [√3 -1 ]

= [√3 +3 ] / [3-√3 ] {multiply nr. and dr. with √3 }

hence (a)

Answer 5

tan 255 = tan 75 = tan(30+45)
=( tan30 +tan 45)/(1-tan30tan45)
=(1/√3 + 1)/(1-1/√3)
=[(1+√3)/√3]/[(√3-1)/√3]
=(1+√3)/(√3-1)
choice b except choice be gives you the wrong sign.