Question

Find the value of tan 13 pie/12

Answer 1

Since 13π1/2 is not an angle where the values of the six trigonometric functions are known, try using the sum or difference formula.13π/12 is not an exact angleFirst, split the angle into two angles where the values of the six trigonometric functions are known. In this case, 13π12 can be split into 3π/4+π/3.tan(3π/4+π/3)Use the sum formula for tangent to simplify the expression. The sum formula for tangent can be derived from the difference formulas for sine and cosine. The derivation of tan(A+B)=sin(A+B)cos(A+B) comes from tan(A+B)=sin(A)cos(B)+cos(A)sin(B)−(cos(A)cos(B)+sin(A)sin(B)).tan(3π/4)+tan(π/3)1−tan(3π/4)tan(π/3)Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.tan(3π/4)=−tan(π/4)The exact value of tan(π4) is 1.tan(3π4)=−1⋅1Multiply −1 by 1 to get −1.tan(3π4)=−1The exact value of tan(π/3) is 3√.tan(π3)=3√Replace the known values in the formula.tan(3π4)+tan(π3)/1−tan(3π4)tan(π3)