tan 15

[mxrider477]

Find the exact value of tan 15 degrees without using tables or a calculator.[/code]

 

[tkhunny]

tan(15°) = tan(30°/2) I suspect there is a formula for those half-angles.

 

[pka]

You are supposed to realize that 15<SUP>o</SUP>=45<SUP>o</SUP>-30<SUP>o</SUP>.
The what does tan(A-B)=?

 

[mxrider477]

ok i know tan 45 equals 1 but i dont know about 30? if i knew that i could figure it out

 

[tkhunny]

tan(30°) = sin(30°)/cos(30°)?

 

[soroban]

Hello, mxrider477!

pka showed you the best way to solve this one.
. . If you want to work out The Long Way, here it is . . .

Find the exact value of tan 15 degrees without using tables or a calculator.

There is a half-angle formula for the tangent.
. . If you don't know it, you can invent it.


We have these identities:
. . sin²(x/2) . = . (1 - cos 2x)/2
. . cos²(x/2) . = . (1 + cos 2x)/2

Then: . sin²(15°) . = . (1 - cos 30°)/2 . = . (1 - (√3)/2)/2 . = . (2 - √3)/4

. . . . . .cos²(15°) . = . (1 + cos 30°)/2 . = . (1 - (√3)/2)/2 . = . 2 + √3)/4

. . . . . . . . . . . . . . . . . sin²15° . . . . .2 - √3
Then: . tan²15° . = . ----------- . = . --------- . = . (2 - √3)²
. . . . . . . . . . . . . . . . . cos²15° . . . . 2 + √3

Therefore: . tan 15° . = . 2 - √3

 

[mxrider477]

thank you guys.