A simple equation involving trigonometry If \cos \theta+\sin \theta+\tan \theta=\sqrt 2+1 then

petrusrexcs

petrusrexcs

Answered question

2022-01-17

A simple equation involving trigonometry
If cosθ+sinθ+tanθ=2+1
then what is the maximum value of
sinθ,sin2θ?

Answer & Explanation

soanooooo40

soanooooo40

Beginner2022-01-18Added 35 answers

Let a=sinθ,b=cosθ. Then a2+b2=1 and
a+b+ab=2+1
Notice that:
a+b|a+b|2(a2+b2)=2
and thus we deduce that ab1. So a,b have the same sign. Since we want the maximum of a, we are only interested in the case when a and b are positive. In this case, using AM-GM:
2+1=a+b+12ab+(112)ab33{a22}+112
and from here, it's easy to find that a12
temzej9

temzej9

Beginner2022-01-19Added 30 answers

If you use the tangent half-angle substitution, you end with
t42t2+4t+11t4=2+1
and one of the roots is t=21. You also have two complex roots and another real which is really messy.

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