How do you use the quadratic formula to solve #3costheta+1=1/costheta# for #0<=theta<360#?

1 Answer
Nghi N
Dec 13, 2016

#64^@26; 140^@14; 219^@86; 295^@74#

Explanation:

#3cos t + 1 = 1/(cos t)#
Cross multiply, and bring the equation to standard form:
#3cos^2 t + cos t - 1 = 0#
Solve this quadratic equation for cos t by using the improved quadratic formula (Socratic Search).
#D = d^2 = b^2 - 4ac = 1 + 12 = 13 #--> #d = +- sqrt13#
There are 2 real roots:
#cos t = -b/(2a) +- d/(2a) = - 1/6 +- sqrt13/6#
#cos t = (- 1 +- sqrt13)/6#

a. #cos t = (- 1 + sqrt13)/6 = 0.434#
Calculator and unit circle give -->
cos t = 0.434 --> arc #t = +- 64^@26#
The co-terminal of t = - 64.26 is #t = 360 - 64.26 = 295^@74#
b. #cos t = (-1 - sqrt13)/6 = - 0.767#
Calculator and unit circle --> arc #t = +- 140^@14#
The co-terminal of t = - 140.14 is #t = 360 - 140.14 = 219^@86#
Answers for (0, 360):
#64^@26; 140^@14; 219^@86; 295^@74#

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