Solve the Problem $C [ 0 , \pi ]$ Have the Inner Product

question 35

Multiple Choice

Solve the problem.
-Let $C [ 0 , \pi ]$ have the inner product $\langle f , g \rangle = \int _ { 0 } ^ { \pi } f ( t ) g ( t ) d t$ , and let $m$ and $n$ be unequal positive integers. Prove that $\cos ( \mathrm { mt } )$ and $\cos ( \mathrm { nt } )$ are orthogonal.

• A) \(\langle \cos ( m t ) , \cos ( n t ) \rangle = \int _ { 0 } ^ { \pi } \cos ( m t ) \cos ( n t ) d t\)
• \[\begin{array} { l }
• = \frac { 1 } { 2 } \int _ { 0 } ^ { \pi } [ \cos ( m t + n t ) + \cos ( m t - n t ) ] d t \\
• = \frac { 1 } { 2 } \left[ \frac { \sin ( m t + n t ) } { m - n } + \frac { \sin ( m t - n t ) } { m + n } \right] \text { from } [ 0 , \pi ] \\
• = 1 .
• \end{array}\]
• B) \[\text { } \begin{aligned}
• \langle \cos ( m t ) , \cos ( n t ) \rangle & = \int _ { 0 } ^ { \pi } \cos ( m t ) \cos ( n t ) d t \\
• & = \int _ { 0 } ^ { \pi } [ \cos ( m t + n t ) + \cos ( m t - n t ) ] d t \\
• & = \left[ \frac { \sin ( m t - n t ) } { m + n } + \frac { \sin ( m t + n t ) } { m - n } \right] \text { from } [ 0 , \pi ] \\
• & = 1 .
• \end{aligned}\]
• C) \[\text { } \begin{aligned}
• \langle \cos ( m t ) , \cos ( n t ) \rangle & = \int _ { 0 } ^ { \pi } \cos ( m t ) \cos ( n t ) d t \\
• & = \frac { 1 } { 2 } \int _ { 0 } ^ { \pi } [ \cos ( m t + n t ) + \cos ( m t - n t ) ] d t \\
• & = \frac { 1 } { 2 } \left[ \frac { \sin ( m t + n t ) } { m + n } + \frac { \sin ( m t - n t ) } { m - n } \right] \text { from } [ 0 , \pi ] \\
• & = 0 .
• \end{aligned}\]
• D) \[\text {} \begin{aligned}
• \langle \cos ( m t ) , \cos ( n t ) \rangle & = \int _ { 0 } ^ { \pi } \cos ( m t ) \cos ( n t ) d t \\
• & = \frac { 1 } { 2 } \int _ { 0 } ^ { \pi } [ \cos ( m t + n t ) - \cos ( m t - n t ) ] d t \\
• & = \frac { 1 } { 2 } \left[ \frac { \sin ( m t + n t ) } { m - n } - \frac { \sin ( m t - n t ) } { m + n } \right] \text { from } [ 0 , \pi ]
• \end{aligned}\]

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Definitions:

Molecular Formula

A notation that indicates the number and type of atoms in a molecule but not their structural arrangement.

Elements of Unsaturation

A measure used in organic chemistry to determine the number of π bonds and rings in a molecule.

Carbon-Carbon Single Bonds

Bonds formed by the sharing of a single pair of electrons between two carbon atoms.

Carbon-Carbon Double Bonds

A chemical bond between two carbon atoms involving four bonding electrons instead of the usual two, leading to the formation of alkenes.