Consider the function \(\displaystyle{f{{\left({x}\right)}}}={\sin{{x}}}+{\cos{{x}}}+{\tan{{x}}}+{\cot{{x}}}+{\sec{{x}}}+{\csc{{x}}}\) Let P be the

Ashleigh Shaffer

Ashleigh Shaffer

Answered question

2022-03-27

Consider the function
f(x)=sinx+cosx+tanx+cotx+secx+cscx
Let P be the smallest possible positive real number such that the equation f(x)=P has real solutions, and let N be the largest possible negative real number such that the equation f(x)=N has real solutions.
Find the value of [1000(P+N)]

Answer & Explanation

Jambrichp2w2

Jambrichp2w2

Beginner2022-03-28Added 12 answers

Hint:
Let sinx+cosx=y2sinxcosx=y21
a=sinx+cosx+tan+cotx+secx+cscx=y+2y21+2yy21
y3ay2+y+a+2=0
Now
2y22y20
Case #1:
If a0,
0=y3ay2+y+a+2222a+2+a+2
a32+2,  but  a0
and
0=y3ay2+y+a+2222a2+a+2
a2320a232

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