Circular Trigonometry Question

Question

Find tan 75° exactly using an angle addition or subtraction formula.

Find the exact value of tan 75°

Answer 1

Tangent subtraction formula is easy to derive from ratio of sine and cosine subtraction formulas.

tan(x-y) = sin(x-y)/cos(x-y) = [sin(x)cos(y)-cos(x)sin(y)]/[cos(x)cos(y)+sin(x)siny)]

dividing numerator and denominator by cos(x)cos(y) we can get tangent subtraction formula

tan(x-y) = [tan(x)-tan(y)]/[1+tan(x)tan(y)]

Let's define x = 135° and y = 60° so

tan(75°) = tan(135°-60°) = [tan(135°)-tan(60°)]/[1+tan(135°)tan(60°)] = [-1-√3]/[1-√3]=(√3+1)²/[(√3-1)(√3+1)]=(3+2√3+1)/((√3)²-1²) = 2+√3

Answer 2

tan(30 + 45) = tan30 + tan45

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1 - tan30tan45

tan(75) = ((\sqrt(3))/(3) + 1

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1 - (\sqrt(3))/(3)

tan(75) = 2 + sqrt(3)

Notes: use knowledge of the unit circle to find the values of tan30 and tan45. (tan = sine/cos = y/x).

use algebra to simplify the expression