If tan A=1/7 and tan B=1/3 find tan(2A + B) without using calculator approximations.
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Step-by-Step Solution
Step 1: Calculate tan(2A)
To find tan(2A+B), we first need to calculate tan(2A). We use the double angle formula for tangent, which states that tan(2A)=1−tan2A2tanA. We are given that tanA=71.
Step 2: Substitute and Simplify tan(2A)
Now, substitute the value of tanA=71 into the formula for tan(2A) and simplify the expression. This involves basic fraction arithmetic.
Step 3: Calculate tan(2A + B)
Now that we have tan(2A) and are given tanB, we can use the sum formula for tangent to find tan(2A+B). The formula is tan(X+Y)=1−tanXtanYtanX+tanY. Here, X=2A and Y=B.
Step 4: Substitute and Simplify tan(2A + B)
Substitute the calculated value of tan(2A)=247 and the given value of tanB=31 into the sum formula. Perform the necessary fraction additions, subtractions, and multiplications to simplify the expression to its final form.