Prove that: 1. Z, Z2 =r,r2[cos(8, + 02) + i sin(8, + 02)] 2.4=(cos(0, -62) + i sin(0, – 0,)] %3D 3. (cos e + i sin 0)" = cos no + i sin no

Q: 2. If z = cos 0 + i sin 0 where 0 is real, show that 1 1 - i tan- = - 1+z 2z 1-z2 Express: а) 1+z2…

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Q: Do the functions y, = cos4t and y, = sin4t constitute a fundamer %3D The functions constitute a…

A: The given differential equation is y''+4y=0 the auxiliary equation is D2+4=0 D2=-4D=±2i therefore…

Q: Prove that: Z, Z2 =r,r2[cos(8, + 02) + i sin(0, + 02)] 2=cos(0, - 62) + i sin(0, – 02)] Z2 3. (cos 0…

A: To prove given statement we need to first convert the polar co ordinate to Euler form then use…

Q: (0, 1) 90° 120 60 (- (-5 ) 150 45 (學學 135 27 2 37 3 4 30 ) 4 6 6. (-1, 0) 180° 0, 2 |(1, 0) 117 6…

A: Given that we need to match the trigonometric functions with  correct  angle measurement in degree.…

Q: Prove that: 1. Z, Z2 =r¡r2[cos(0, + 02) + i sin(0, + 02)]

A: According to our company policy we can only answer first question and rest of questions can be…

Q: Prove using the IVT cosx = cos-1x has a solution in (0, 1).

A: cos x = x   rearrange to x - cos x = 0

Q: Given that Y1 = x-1/2 cos x and y2 = x-1/2 sin x from a fundamental set of solutions of (--)- x²y" +…

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Q: snip

A: Given, cosjx,coskx, ∋j≠k, k≥1 if inner product of two function f(x), g(x) on the interval [a,b]…

Q: 9. Find 14 values of S and R obtained from the difference equations: Sk41 = R, (1/cos(0.3S, )) %3D…

A: So this question is to be solved using Matlab code: I will share the input and output for this…

Q: Q3-Prove that (cos303cos³ 0-3cos 0 sin? 0)

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Q: 2. : (a) For n ≥ 1, show that 1+z+2²+...+ 2¹: (b) Hence, show that 1+ cos 0 + cos 20 + 12+1 2 #1 7…

A: Solution:

Q: Prove that: 2n 1 cos=0 + i sin 0) + 10 2cosne - %3D eino

A: If  is an exponential form of a complex number where x and y are real numbers then its polar form…

Q: 2. Let f(r, y,z) = In(r² + 2) + ye* – cos(2yz). Find faya- COS

A: see below the answer as per bartleby guideline we have to solve the first question onl

Q: 3. Show that cos 30 = cos 0 – 3 cos 0. [Hint: Apply De Moivre's Theorem.] |

A: Given: verify that cos3θ=cos3θ-3cosθ according to given hint theorem cosx+isinxn=cosnx+isinnx

Q: 1 Prove that sec 03D Cos 0

A: Let P(x,y) be a point on a circle of radius r associated with the angle θ.

Q: 1) Prove that : [cos 0 – i sin 0] 2) For z = -4 + 4i , find z in exponension form. ||

A: Respected student, By the norms of Bartleby we are solving your first two questions, if you need a…

Q: 14. If we let u = 4x – 1 for the (2tan(4x-1)-3 cos“(4x=1) dx. what will be the cos(4x-1) resulting…

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Q: Let z = 8(cos 50° + i sin 50°) and w=2(cos 30° + i sin 30°). Express in the form r(cos+ i sin 0):…

A: Here we can use the following formula to find the results eiθ=cos θ+i sinθande-iθ=cos θ-i sinθ

Q: Z1 "(cos(0, r2 Z2 02) + isin(0, – 02)](z2 # 0). 85. Let z1 = ri(cos 0, + isin 01) and z2 = r2(cos 0,…

A: Given z1=r1cosθ1+isinθ1 and z2=r2cosθ2+isinθ2.

Q: Q.3- Prove that: 2n 1 (cos =0 +i sin-) + 10 2cosne eino

A: The given problem is to prove the following complex identity. (cos n/5 θ +i sin 2n/10 θ)+1/einθ =…

Q: 8. Show that cos a – cos y| < |x – y| for all æ and y.

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Q: A force of F(x)=x2−cos(3x)+2, x is in meters, acts on an object. What is the work required to move…

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Q: B- Find the solution to the equation = k satisfying the conditions at ax2 1-- 30 at x 0 for all…

A: A

Q: Solve 2cos? (x) +3 cos (x)+1=0 for all solutíons osx<2 TT

A: Given 2cos2x+3cosx+1=00≤x≤2π

Q: 4. Let z1 = 24(cos(75°) + i•sin(75°)) and z2 = 2(cos(15°) + i • sin(15°) ). Compute the %3D %3D…

A: SOLUTION-

Q: Prove that there exists a value of x so that -1 COS T = cosx %3D ned to

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Q: Compute | cos z dz. cos z +C 3 cos a + sin z +C 3 sin r + cos x +C 3 Cos x - sin I sin z + 5 sin x…

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Q: Q1: show that1) d/dz [cosz] = -sin z , 2) * 1+tan2 z = sec2 z

A: In both solutions i have used the definition

Q: 18.) Suppose x = R cos(0) sin($), y = R sin(0) sin($) and z = x² + y? + z? = R²Y R cos(). Verify the…

A: Given

Q: A force of F(x)=x2−cos(3x)+2, x is in meters, acts on an object. What is the work required to move…

A:

Q: 2-If w(x, t) = Sin(x - + 3t) + Cos(2x+ 6t), prove that wtt = 9wxx %3D %3D

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Q: Evaluate the integral. 1/2 |[cos 9t i- 3 sin tj + sin? 3t k] dt T/2 | (cos 9t i- 3 sin tj+ sin 3t k]…

A: We have function ∫0π2r(t) =∫0π2[cos9t i -3sint j + sin23t k ]dt

Q: 5- Prove the following (Answer two only): secA. secB.cscA.cscB a- cos*e – sin*o =cos²0 – sin²e c- =…

A: Answer 2 only.

Q: Let = r, (cos 0, + isin 0, ) and r, (cos 0, + i sin 0, ). Z, = X1 + iy, 1 Z 2 X2 + iy, Then show…

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Q: 4/ Prove that: " cos bx dx = (bsinbx +a cos bx]+c

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Q: 19. For all positive angles less than 360°, if csc(2x +30°) = cos(3y- 15°), the %3D sum of x and y…

A: Given that csc(2x+30°)=cos(3y-15°) It is known that csc(x)=1sinx. So, 1=sin(2x+30°)cos(3y-15°)…

Q: (a) Prove that dv 1 2 u? + v2 arctan u- + C. 2 u 2, /2/2 ru (b) Prove that I dv du 2 - u? + v2 18 by…

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Q: 0.3- Prove that: n 2n 1 (cos =0 +i sin8) + 10 2cosne - 5 eine

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Q: Prove that (a) Pn (cos 0) = 2F1(-n,n+ 1; 1; sin²0/2).

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Q: 9 If f(0) = cos 0, cos 02 cos 03 ..... cos 0n %3D Show, {tan 01 + tan 02 + tan 03 + ... + tan 0,} =…

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Q: Express Q(x) = cos4 (x )+ 12 as the composition of three func-tions; that is, identify f, g, and h…

A: Q(x) = cos4 (x )+ 12 We need to express the given function as a composition of three functions  that…

Q: 2n 1 (cos=0 0 +i sin e) + 10 2cosno eine

A: The general form of a complex number is given by z=a+ib, where a,b are two real numbers and i=-1.…

Q: a. Show that W[5, sin2 t, cos(2t)] = 0 for all t by directly evaluating the Wrosnkian.

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Q: Prove that: Z,Z2 =r,r2[cos(0, + 02) + i sin(0, + 02)] 1.

A: NOTE- As we are allowed to answer one question in one session so, i have answered first one for you.…

Q: 22-j is 1. -1+jZ If Z = cos 0 + j(1+ sin 0), show that the magnitude of

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Q: 2. Prove that cos30 = (4cos'e - 3cos0) and sin30 = (3sine – 4sin e) using De Moivre's Theorem. %3D…

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Q: (d) Show that there is some x with 0 < x < 2 such that x2 + cos(ax) = 4. %3D

A: Solution-

Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.