Courses
Courses for Kids
Free study material
Offline Centres
More
Store Icon
Store

Show that max and min values of $8\cos \theta - 15\sin \theta$ are 17 and -17 respectively.

seo-qna
Last updated date: 04th Mar 2024
Total views: 21.6k
Views today: 1.21k
IVSAT 2024
Answer
VerifiedVerified
21.6k+ views
Hint: Here, we will use the extreme values of the form $a\cos \theta + b\sin \theta
 $ to find the max and min values.

Given,
$8\cos \theta - 15\sin \theta \to (1)$
Let us compare the equation (1) with $a\cos \theta + b\sin \theta $, we get
$a = 8,b = - 15$
As, we know the maximum and minimum values of $a\cos \theta + b\sin \theta $ are $\sqrt
 {{a^2} + {b^2}} $ and -$\sqrt {{a^2} + {b^2}} $respectively.
Therefore, substituting the values of a and b, we get
$
   \Rightarrow \max = \sqrt {{a^2} + {b^2}} = \sqrt {{8^2} + {{( - 15)}^2}} = \sqrt {64 + 225} =
 \sqrt {289} = 17 \\
   \Rightarrow \min = - \sqrt {{a^2} + {b^2}} = - \sqrt {{8^2} + {{( - 15)}^2}} = - \sqrt {64 + 225} = - \sqrt {289} = - 17 \\
$
Hence, the maximum value of $8\cos \theta - 15\sin \theta$ is 17 and minimum value of
 $8\cos \theta - 15\sin \theta$ is -17.
Note: The maximum and minimum of the $a\cos \theta + b\sin \theta $ will differ only by
 the sign of the value i.e.., the maximum value will have the positive sign whereas the minimum value will have the negative sign of the same value.