Question

Find the exact value of the expression tan(-165) using the half-angle formula.

Answer 1

angle is in 3rd quadrant, tan is equal in 1st and 3rd quadrants
\[\tan(-165) = \tan(195) = \tan(15) = \tan(\frac{30}{2})\]
half-angle identity is:
\[\tan \frac{\theta}{2} = \frac{1-\cos \theta}{\sin \theta}\]

Answer 2

Is tan195 a coterminal angle of tan(-165)?

Answer 3

yes

Answer 4

Okay, so how did you get to 15 from 195?

Answer 5

reference angles

Answer 6

And also, I tried to work it out myself and made it to \[\pm \sqrt{\frac{ 1+(\frac{ -\sqrt{3} }{ 2 }) }{ \frac{ 1 }{ 2 } }}\] But I don't know how to get to the answer from there (if that is even correct) of 2-\[\sqrt{3}\]

Answer 7

\[2-\sqrt{3}\]

Answer 8

Okay, the reference angle part makes sense :)

Answer 9

yes that is correct

Answer 10

Do you know how or can you tell me how I can get to \[2-\sqrt{3}\] from that point?

Answer 11

oh sorry i misread ... i meant answer is correct

well there are different half-angle identities you can use
\[\tan \frac{x}{2} = \sqrt{\frac{1-\cos x}{1+\cos x}} = \frac{1-\cos x}{\sin x} = \frac{\sin x}{1+\cos x}\]

Answer 12

that okay, thank you :)