Bring the quadratic equation in cot x to standard form:
cot^2 x + 4cot x - 7 = 0cot2x+4cotx−7=0
Use the new improved quadratic formula (Socratic Search)
D = d^2 = b^2 - 4ac = 16 + 28 = 44D=d2=b2−4ac=16+28=44 --> d = +- 2sqrt11d=±2√11
There are 2 real root:
cot x = -b/(2a) +- d/(2a) = -4/2 +- 2sqrt11/2 = -2 +- sqrt11cotx=−b2a±d2a=−42±2√112=−2±√11
cot x1 = -2 + sqrt11 = 1.32cotx1=−2+√11=1.32,
cot x2 = -2 - sqrt11 = - 5/32cotx2=−2−√11=−532
Use calculator and unit circle -->
tan x1 = 1/1.32 = 0.7575tanx1=11.32=0.7575 --> Two values of x1 -->
x1 = 37^@15x1=37∘15, and x1 = 37.15 + 180 = 217^@15x1=37.15+180=217∘15
tan x2 = 1/-5.32 = - 0.19tanx2=1−5.32=−0.19 --> Two values of x2-->
x2 = 10^@65x2=10∘65, and x2 = 10.65 + 180 = 190^@65x2=10.65+180=190∘65
