Find all solutions in the interval [0,2PI)
- Thread starter kdemmerle
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It had me going there for a minute, but I finally saw it. Easier than you might think.
25 sinx cosx - 5sinx + 20cosx = 4
25sin(x)cos(x) - 5sin(x) + 20cos(x) - 4 = 0
25sin(x)cos(x) + 20cos(x) - 5sin(x) - 4 = 0
5*cos(x)*[5*sin(x) + 4] - (5*sin(x) + 4) = 0
[5*cos(x) - 1]*[5*sin(x) + 4] = 0
Can you take it from there?
25 sinx cosx - 5sinx + 20cosx = 4
25sin(x)cos(x) - 5sin(x) + 20cos(x) - 4 = 0
25sin(x)cos(x) + 20cos(x) - 5sin(x) - 4 = 0
5*cos(x)*[5*sin(x) + 4] - (5*sin(x) + 4) = 0
[5*cos(x) - 1]*[5*sin(x) + 4] = 0
Can you take it from there?
Thank you for your help. Actually, I never took Trig, so I am basically trying to teach myself in order to help my husband in his Contemporary Precalculus class. You have helped me tremendously, but I don't fully understand how to find all of the solutions.
Kris
Kris
- Joined
- Apr 12, 2005
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- 11,339
Third-party tutoring rarely is very effective.
Interestingly, we haven't done any Trig, yet. This is just algebra all the way through the next step.
We exploit this principle, if A*B = 0, the either A = 0 or B = 0.
Given this equation
[5*cos(x) - 1]*[5*sin(x) + 4] = 0
5*cos(x) - 1 = 0 or 5*sin(x) + 4 = 0
This leads to
cos(x) = 1/5 or sin(x) = -4/5
FINALLY, we get to some trigonometry! Each of these has two solutions on [0,2*pi]. They are not pretty answers. I suspect you'll need a calculator. Make sure it's set for radians. The calculator will give you only one answer for each. finding the other is your challenge for the day.
Interestingly, we haven't done any Trig, yet. This is just algebra all the way through the next step.
We exploit this principle, if A*B = 0, the either A = 0 or B = 0.
Given this equation
[5*cos(x) - 1]*[5*sin(x) + 4] = 0
5*cos(x) - 1 = 0 or 5*sin(x) + 4 = 0
This leads to
cos(x) = 1/5 or sin(x) = -4/5
FINALLY, we get to some trigonometry! Each of these has two solutions on [0,2*pi]. They are not pretty answers. I suspect you'll need a calculator. Make sure it's set for radians. The calculator will give you only one answer for each. finding the other is your challenge for the day.