Math

What are the steps to getting the tan of 345degree, without the use of a calculator?

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  1. 345°=360°-15°

    tan(A-B)=(tan A - tan B)/(1 - (tan A)*(tan B))

    tan(360°)=0

    tan(360°-15°)= tan(360°)-tan(15°)/(1-(tan(360°)*(tan(15°))

    tan(360°-15°)= 0-tan(15°)/(1-0*(tan(15°))

    tan(360°-15°)=tan(345°)= -tan(15°)/(1-0)

    tan(345°)= -tan(15°)/1

    tan(345°)= -tan(15°)

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  2. tan(x/2) = [1 - cos(x)]/sin(x)

    tan(30°/2)=tan(15°)

    sin(30°)= 1/2

    cos(30°)= sqroot(3)/2

    tan(30°/2)=tan(15°)= [1 - cos(30°)]/sin(30°)
    = (1 - sqroot(3)/2)/(1/2)
    = 2 - sqroot(3)

    tan(15°) = 2 - sqroot(3)

    tan(345°)= -tan(15°)

    tan(345°)= -(2 - sqroot(3))

    tan(345°)= sqroot(3)-2

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  3. or ...

    345° is coterminal with -15°
    so tan 345° = - tan15
    = - tan(45-30)
    = - (tan45 - tan30)/(1 + tan45tan30)
    = -(1-1/√3)/(1+1/√3) , multiply top and bottom by √3
    = (1 - √3)(√3 + 1)

    You can rationalize if necessary to get Bosnian's answer

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