Find the least value of \(\displaystyle{4}{{\csc}^{{2}}{x}}+{9}{{\sin}^{{2}}{x}}\)

Briley Cabrera

Briley Cabrera

Answered question

2022-04-13

Find the least value of 4csc2x+9sin2x

Answer & Explanation

webhui2v2

webhui2v2

Beginner2022-04-14Added 10 answers

(2sinx3sinx)20
csc2x+9sin2x12
with equality achieved when sin2x=23
chambasos6

chambasos6

Beginner2022-04-15Added 12 answers

Let y=4csc2x+9sin2x
y=8csc2xcotx+9sin2x
y=8csc2x(csc2x)8cotx(2csc2xcotx)+18cos2x
y=8csc4x+16csc2cot2x+18cos2x
for maxima or minima, y=0 hence,
8csc2xcotx+9sin2x=0
2cosx(9sinx4sin3x)=0
cosxsin3x(9sin4x4)=0
cosx=0x=π2
or 9sin4x4=0sin2x=23
One, can easily check that minimum of y occurs for sin2x=23(y>0), hence, substituting sin2x=23 in y, the minimum value is
ymin=4sin2x+9sin2x=423+9(23)=12

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?