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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