How do you use the quadratic formula to solve 3costheta+1=1/costheta3cosθ+1=1cosθ for 0<=theta<3600≤θ<360?

1 Answer
Nghi N
Dec 13, 2016

64^@26; 140^@14; 219^@86; 295^@7464∘26;140∘14;219∘86;295∘74

Explanation:

3cos t + 1 = 1/(cos t)3cost+1=1cost
Cross multiply, and bring the equation to standard form:
3cos^2 t + cos t - 1 = 03cos2t+cost−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=d2=b2−4ac=1+12=13--> d = +- sqrt13d=±√13
There are 2 real roots:
cos t = -b/(2a) +- d/(2a) = - 1/6 +- sqrt13/6cost=−b2a±d2a=−16±√136
cos t = (- 1 +- sqrt13)/6cost=−1±√136

a. cos t = (- 1 + sqrt13)/6 = 0.434cost=−1+√136=0.434
Calculator and unit circle give -->
cos t = 0.434 --> arc t = +- 64^@26t=±64∘26
The co-terminal of t = - 64.26 is t = 360 - 64.26 = 295^@74t=360−64.26=295∘74
b. cos t = (-1 - sqrt13)/6 = - 0.767cost=−1−√136=−0.767
Calculator and unit circle --> arc t = +- 140^@14t=±140∘14
The co-terminal of t = - 140.14 is t = 360 - 140.14 = 219^@86t=360−140.14=219∘86
Answers for (0, 360):
64^@26; 140^@14; 219^@86; 295^@7464∘26;140∘14;219∘86;295∘74

Related questions
See all questions in Solving Trigonometric Equations
Impact of this question
1575 views around the world
You can reuse this answer
Creative Commons License