Answer
Verified
91.5k+ 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.
$ 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.
Recently Updated Pages
Name the scale on which the destructive energy of an class 11 physics JEE_Main
Write an article on the need and importance of sports class 10 english JEE_Main
Choose the exact meaning of the given idiomphrase The class 9 english JEE_Main
Choose the one which best expresses the meaning of class 9 english JEE_Main
What does a hydrometer consist of A A cylindrical stem class 9 physics JEE_Main
A motorcyclist of mass m is to negotiate a curve of class 9 physics JEE_Main
Other Pages
Derive an expression for maximum speed of a car on class 11 physics JEE_Main
If a wire of resistance R is stretched to double of class 12 physics JEE_Main
3 mole of gas X and 2 moles of gas Y enters from the class 11 physics JEE_Main
The vapour pressure of pure A is 10 torr and at the class 12 chemistry JEE_Main
Electric field due to uniformly charged sphere class 12 physics JEE_Main
If the central portion of a convex lens is wrapped class 12 physics JEE_Main