2n The space is C[0,2r] and the inner product is (f.g) = f(t)g(t) dt. Show that sin mt and cos nt are orthogonal for all positive integers m and n. Begin by writing the inner product using the given functions. (sin mt, cos nt) = dt Use a trigonometric identity to write the integrand as a sum of sines. 2x 1 (sin mt, cos nt) =- dt Then integrate with respect to t. (sin mt, cos nt) = Evaluate the result at the end points of the interval. Note that m -n in the denominator means that this result does not apply to m = n. (sin mt, cos nt) = [(O - 0) (Simplify your answers.) Then simplify this result to get the inner product for all positive integers m # n. (sin mt, cos nt) =
2n The space is C[0,2r] and the inner product is (f.g) = f(t)g(t) dt. Show that sin mt and cos nt are orthogonal for all positive integers m and n. Begin by writing the inner product using the given functions. (sin mt, cos nt) = dt Use a trigonometric identity to write the integrand as a sum of sines. 2x 1 (sin mt, cos nt) =- dt Then integrate with respect to t. (sin mt, cos nt) = Evaluate the result at the end points of the interval. Note that m -n in the denominator means that this result does not apply to m = n. (sin mt, cos nt) = [(O - 0) (Simplify your answers.) Then simplify this result to get the inner product for all positive integers m # n. (sin mt, cos nt) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Write the inner product for the case m =n and integrate.
2n
| sin mt cos mt dt =
Evaluate the integral.
2n
sin mt cos mt dt =
(Simplify your answer.)
Therefore, sin mt and cos nt are orthogonal for all positive integers m and n because the inner product is always
![2n
The space is C[0,2r] and the inner product is (f.g) =
f(t)g(t) dt. Show that sin mt and cos nt are orthogonal for all positive integers m and n.
Begin by writing the inner product using the given functions.
(sin mt, cos nt) =
dt
Use a trigonometric identity to write the integrand as a sum of sines.
2x
1
(sin mt, cos nt) =-
dt
Then integrate with respect to t.
(sin mt, cos nt) =
Evaluate the result at the end points of the interval. Note that m -n in the denominator means that this result does not apply to m = n.
(sin mt, cos nt) = [(O - 0)
(Simplify your answers.)
Then simplify this result to get the inner product for all positive integers m # n.
(sin mt, cos nt) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc5941ae9-6ead-4355-86dd-455bc2b61a77%2F6f209c68-1021-4d6e-a129-2a02bd0a8b4f%2Fwu1gyr9_processed.png&w=3840&q=75)
Transcribed Image Text:2n
The space is C[0,2r] and the inner product is (f.g) =
f(t)g(t) dt. Show that sin mt and cos nt are orthogonal for all positive integers m and n.
Begin by writing the inner product using the given functions.
(sin mt, cos nt) =
dt
Use a trigonometric identity to write the integrand as a sum of sines.
2x
1
(sin mt, cos nt) =-
dt
Then integrate with respect to t.
(sin mt, cos nt) =
Evaluate the result at the end points of the interval. Note that m -n in the denominator means that this result does not apply to m = n.
(sin mt, cos nt) = [(O - 0)
(Simplify your answers.)
Then simplify this result to get the inner product for all positive integers m # n.
(sin mt, cos nt) =
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