Trig Identity Question!Watch

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#1
Without using a calculator, find the value of:

1 - tan 15degrees
1 + tan 15degrees

.. I took the tan 15 degrees as tan (45 - 30) degrees and then tried the addition formula for tan(A - B) but then it goes all complicated and i can't cancel down. Is that the right method and i'm just going wrong somewhere or am i completely off track

The answer is root 3 over 3 .. if it'll help anyone get it
0
13 years ago
#2
(Original post by violet)
Without using a calculator, find the value of:

1 - tan 15degrees
1 + tan 15degrees

.. I took the tan 15 degrees as tan (45 - 30) degrees and then tried the addition formula for tan(A - B) but then it goes all complicated and i can't cancel down. Is that the right method and i'm just going wrong somewhere or am i completely off track

The answer is root 3 over 3 .. if it'll help anyone get it
You can get the exact value of tan 15 like this:

Draw equilateral triangle ABC with sides of 2 units
Drop a perpendicular from the apex B to the mid-point of the base.(M)
Produce MB vertically upwards to 2 units above B. Call this point D.
DBC is an isosceles triangle with angles 15, 15, 150.
Tan 15 (angle BDC) is then 1/(2+√3)

The rest should be a lot of fiddly stuff with surds.

Aitch
0
13 years ago
#3
I think using the formula:
tan( A+/- B) = (tanA +/- tanB)/( 1 -/+ tanAtanB)

But it's complicated at all. So I think of another solution:
Using sin2A = 2tanA/(1+tan^2A) and cos2A = (1- tan^2A)/(1+tan^2A)

I'm not sure the formulae are written..correctly or not.
0
13 years ago
#4
(Original post by Aitch)
You can get the exact value of tan 15 like this:

Draw equilateral triangle ABC with sides of 2 units
Drop a perpendicular from the apex B to the mid-point of the base.(M)
Produce MB vertically upwards to 2 units above B. Call this D.
DBC is an isosceles triangle with angles 15, 15, 150.
Tan 15 (angle BDC) is then 1/(2+√3)

The rest should be a lot of fiddly stuff with surds.

Aitch
It is!

1/(2+√3)+1 = (3+√3)/(2+√3) (1)
1/(2+√3)-1 = (1+√3)/(2+√3) (2)

(2)/(1) = (1+√3)/(3+√3) = (2.√3)/6 = (√3)/3 or 1/√3

Aitch
0
#5
Cleverclogs Thanks everyone!
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