67. Use Equation (4) to write the following powers of e in the form a + bi, a. eir b. e/4 c. e/2

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A: Using the formulas of coshx and sinhx

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Q: find the four non zero terms in the maclaurin serise for sin h-lx

A: Formula Used: We will need following binomial expansion:

Q: If n is an odd natural number prove that I =, xsinnx dx = sinx 2.

A: Consider the given: It is given that n is an odd natural number, Take n = 1,Substituting n=1 and…

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A: Givenr=4-u2sinuTo finddrdu

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A: we have to find the derivative of y

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A: If, X=B1sinωt and Y=B2sinωt-θ, then, it can be said that Y lags X by an angle θ .

Q: 39. Write 16cos*x in terms of first powers of cosine.

A: Given           16cos4x

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Q: 3. e<4TI. 2. ind 3. where when tan e = -

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Q: In the given question, verify the identity:-   cosh2 x =- 1 + cosh 2x/ 2

A: To verify: cosh2x=-1+cosh 2x2

Q: 2. Make the substitution x = 5 tan(0) into the expression (25 + x²)3/2 and simplify. Leave your %3D…

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Q: (-) - Verify the identity. = -sin e 2

A: To verify the identity: cos3π2-θ=-sinθ Verification: Simplifying the left hand side.…

Q: 1 dr r/1+ x6 (use trigonometric substitution) 3.

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Q: 87 cos e de 4 O 0 O 6

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Q: |2sin*e cos e de= .a 315 8. .b O 63 16 .cO 315 16 60 00

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Q: Express in the form a + bi where a and b are real numbers.

A: To: Express in the form a + bi where a and b are real numbers 5(cosπ+i sinπ)

Q: 5. Show that de (tan 8) can be expressed in the form 3tan 8+3sec2e-3. Hence, find tan*e de.

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Q: Show that (1 – e") = 2 sin ( - a eil(a-r)/2]

A: To show 1-ejα=2 sinα2ejα-π2 , simplify left hand side expression and show equivalent to right hand…

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A: Given that ∫5 sin θ cos θsin2θ + 3 sin θ-4dθ We can solve this by using substitution method. Needed…

Q: what is Selution of DE (D-20)Y=ê Sinx When Sotved using the meth ed of undetermind

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Q: be +50 + 10 cos z)dr= %3D

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Q: 3. In (cos)

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Q: de 4TV3 Show that (2+cos8) 9

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Q: claim you make. 00 1 tan n² n=1 (a)

A: Using the limit comparison test.

Q: Express the following in complex exponential form: /cos(y) + jsin(y)] Where x = 2 and y = 142. %3D…

A: Given that: x=2 and y=142, x[cos(y°)+jsin(y°)]

Q: (b) у — 2г tan-"(г) — In(1 + x?)

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Q: Evaluate: fa 3tan e de cos20

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Q: T/4 (b) cos x dx - 1/4

A: Since you have asked multiple questions in single request  so we  will be answering only first…

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A: The complex mathematical expressions can be solved using calculators.

Q: O cos 70 do

A: To solve the following integral

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A: Suppose a cell is suspended in a solution containing a solute of constant concentration Cs. Suppose…

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A: In a polar coordinate system, a point is represented with coordinates (r,θ). The radial coordinate r…

Q: dy' By 34?-5 homogenaus equa tion: -3u %3D

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Q: Express sinh (i) in the form a + ib

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Q: 17 8 15 What is the sin(ZG)?

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Q: S St. Not e (cos48)² 45 ds

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Q: 1 de 4-cose EIN

A: I have provided solution in step2.

Q: 6. Solve the following difterence equatrm using z-T. b. yck)- y(k-1) +yCk-2) = 2 sin H -yck-2)=2sin…

A: Given y(k)-34y(k-1)+18y(k-2)=2sinkπ2               (1) Since Zy(k-1)=y(-1)+z-1Y(z)…