C4 trig question

Given that tan 45º = 1, use the identity for tan2x to show that the exact value of tan22.5º = √2 - 1. So far, I've done:

2 tan22.5 / 1 - tan^2(22.5) = 1
2 tan22.5 = 1 - tan^2(22.5)

But I'm not entirely sure where I'm going...?

Reply 1

veni, vidi, vici

11

2tan(22.5)= 1- tan^2(22.5)

rearrange as a quadratic,

tan^2(22.5) + 2tan(22.5) -1 =0

use -b+/- *squareroot* b^2-4ac , all over 2a

i don't know if this is a flawed method but :smile:

Reply 2

El Matematico

tan2A = (2tanA)/(1-tan^2A)

So, set A = 22.5, then tan45 = (2tan22.5)/(1-tan^222.5)
=> (1-tan^2(22.5))/2 = tan22.5 (Rearranging)
=>(1 - (sec^2(22.5) - 1)/2 = tan22.5 (Using trig identity for tan^2x)

=>(2 - sec^2(22.5))/2 = tan22.5 (*)

Now use cos2x = 2cos^2x - 1
=> sec^2x = 2/(cos2x + 1)
Substitute into (*), giving

(2 - 2/(cos45 + 1))/2 = (2 - 2/(sqrt(2)/2 + 1))/2

Go from there and the answer should come out.

Reply 3

Creole

2

I believe you are on the right track. Just bring all terms to one side and you have a quadratic.

However i get two answers, only one of which is √2 - 1, the other is -√2 - 1.

Reply 4

TheDuck

2

Excalibur

Given that tan 45º = 1, use the identity for tan2x to show that the exact value of tan22.5º = √2 - 1. So far, I've done:

2 tan22.5 / 1 - tan^2(22.5) = 1
2 tan22.5 = 1 - tan^2(22.5)

But I'm not entirely sure where I'm going...?

so now you have a quadratic with x=tan(22.5)

x²+2x-1=0
so (x+1)²=2
so x+1=±√2
so x=±√2-1

Reply 5

El Matematico

Using that its a quadratic is easier and is better than hitting it with identities like I did. Go with that method although Ill leave mine posted in case youre interested.

Reply 6

Excalibur

OP

15

Oooh, I see. :smile:

Thanks everyone.

C4 trig question

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