http://prntscr.com/fs8e38 - question
http://prntscr.com/fs8dxe - answer
Since they didn't give a domain for theta, should the answer be plus or minus x/9sqrt(9-x^2) + C ?
Thanks
2016 HSC (Accelerated)
// 2 Unit Maths // Studies of Religion 1 //
2017 HSC
// Biology // Physics // Maths Extension 1 // Maths Extension 2 // English Advanced //
They give x = 3sintheta, so we don't know if it's the first or second quadrant so if it's the second quadrant then tantheta is negative, so just wondering if it would be plus or minus.
EDIT: http://prntscr.com/fs8ru9
From this question they drew a triangle with plus or minus which is where I got the idea from
Last edited by pikachu975; 6 Jul 2017 at 6:07 PM.
2016 HSC (Accelerated)
// 2 Unit Maths // Studies of Religion 1 //
2017 HSC
// Biology // Physics // Maths Extension 1 // Maths Extension 2 // English Advanced //
Funny Hahah just did this question in a CSSA paper.
Getting from the second last step to the last step you must construct a triangle and hence your theeta is less than 90. This automatically puts it in the first quadrant and hence your answer is always positive.
2016 HSC (Accelerated)
// 2 Unit Maths // Studies of Religion 1 //
2017 HSC
// Biology // Physics // Maths Extension 1 // Maths Extension 2 // English Advanced //
As a quick way of showing why the antiderivative cannot be minus x/(9sqrt(9-x^2)) + C, try differentiating this expression and see if you get the original expression in the question.
The answer for this question is plus x/(9sqrt(9-x^2)) + C because they assumed that costheta = plus sqrt (1-sin^2 theta) when moving from line 1 to 2, and implicitly assumed this again when moving from line 4 to 5. If they assumed that cos theta = minus sqrt(1-sin^2theta), then the minuses would've cancelled out since they use this assumption in moving from line 1 to 2, then again from line 4 to 5. So in the end, your final expression for the antiderivative would be the plus version.
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