Get 2 FREE Instant-Explanations on Filo with code FILOAPP

Find the set of values of parameter a so that the equation (si

Question

Find the set of values of parameter a so that the equation $(s i n^{- 1} x)^{3} + (co s^{- 1} x)^{3} = a π^{2}$ has a solution

Views: 753 students

Found 5 tutors discussing this question

Amelia Discussed

Find the set of values of parameter a so that the equation

(s i n^{- 1} x)^{3} + (co s^{- 1} x)^{3} = a π^{2}

has a solution

15 mins ago

Discuss this question LIVE

15 mins ago

Text solutionVerified

Let

y = (sin^{- 1} x)^{3} + (cos^{- 1} x)^{3} = (sin^{- 1} x + cos^{- 1} x) [(sin^{- 1} x)^{2} + (cos^{- 1} x)^{2} - sin^{- 1} x . cos^{- 1} x]

\Rightarrow y = \frac{π}{2} [(sin^{1 -} x + cos^{- 1} x)^{2} - 3 sin^{- 1} x . cos^{- 1} x]

\Rightarrow y = \frac{π}{2} [\frac{π ^{2}}{4} - 3 sin^{- 1} x (\frac{π}{2} - sin^{- 1} x)]

\Rightarrow y = \frac{π}{2} ⎣ ⎡ \frac{π ^{2}}{16} + 3 (\frac{π}{4} - sin^{- 1} x)^{2} ⎦ ⎤

Now we know,

- \frac{π}{2} \leq sin^{- 1} x \leq \frac{π}{2}

Thus range of y is

[\frac{π ^{3}}{32}, \frac{7 π ^{3}}{8}]

Thus given equation to have a solution

a \in [\frac{1}{32}, \frac{7}{8}]

Was this solution helpful?

150

Report

One destination to cover all your homework and assignment needs

Learn Practice Revision Succeed

Instant 1:1 help, 24x7

60, 000+ Expert tutors

Textbook solutions

Big idea maths, McGraw-Hill Education etc

Essay review

Get expert feedback on your essay

Schedule classes

High dosage tutoring from Dedicated 3 experts

Trusted by 4 million+ students

Stuck on the question or explanation?

Connect with our maths tutors online and get step by step solution of this question.

231 students are taking LIVE classes

Question Text	Find the set of values of parameter a so that the equation $(s i n^{- 1} x)^{3} + (co s^{- 1} x)^{3} = a π^{2}$ has a solution
Answer Type	Text solution:1
Upvotes	150