Sponsor Area

Principle Of Mathematical Induction

Question

Find the maximum and minimum values of:

$WiredFaculty$

Solution

Let $space space straight u space equals space 4 space cosα space space minus space 3 space sinα$
Put $4 space equals space straight r space cosβ comma space space space space 3 space equals space straight r space sinβ$
Squaring and adding, we get
$straight r squared left parenthesis cos squared straight beta plus sin squared straight beta right parenthesis space equals space 16 plus 9 equals 25$
$rightwards double arrow$    $space space straight r squared space equals space 25$ or r = 5
∴ $straight u equals straight r space cosβ space cosα space minus space straight r space sinβ space sinα space equals space straight r space cos left parenthesis straight beta plus straight alpha right parenthesis space equals space 5 space cos left parenthesis straight beta plus straight alpha right parenthesis$
Now, $negative 1 less or equal than cos left parenthesis straight beta plus straight alpha right parenthesis less or equal than 1$
$rightwards double arrow$    $negative 5 less or equal than 5 cos left parenthesis straight beta plus straight alpha right parenthesis less or equal than 5$
$rightwards double arrow$    $negative 5 less or equal than straight u less or equal than 5$
$rightwards double arrow$    $space space space space space space space space space space space space space space space space space space space space space space space space space space minus 5 less or equal than 4 cosα space minus space 3 space sinα less or equal than 5$
Hence, max. value of $open curly brackets 4 space cosα space minus space 3 space sinα close curly brackets space equals space 5$
Min. value of $WiredFaculty$