3. Give upper and lower estimates (in terms of n alone) for sin nb COs na + valid for all a, b; show these estimates are the best possible, by giving values of a and b for which the bounds are actually attained.

Q: 20 a Show that sin (A + B) + sin (A – B) = 2 sin A cos B. b Hence find: i 2 sin 3x cos 2x dx ii sin…

A:

Q: 5. (a) Expand (1+x) and hence find the value of (1.02) correct to 4 decimal place 1+sin 0 m-1 1-sin…

A: First expand 1+x4 then find the value of 1.044 Using the identity, a+b4=a4+4a3b+6a2b2+4ab3+b4…

Q: 1. Determine the absolute extrema of f(x) = (sin r – 1) cos r on 2 2 if they exist, and identify the…

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Q: The integral of the given function | cos(9x) cos(5x)dx is equal .. 14 sin 4x + 4 sin 14x А. · + C 12…

A: This question can be solved by applying the formula of 2.cosA.cosB

Q: 7) Use Theorem 7.1.1 (Table) to find L{f (t)} for the following: F()=cos(V2 t)+ sin(3:) n(3t) b. =…

A: Given ft=cos2t+sin3t

Q: The approximation of 1 = f*cos (x5)dx using composite Simpson's rule with n = 3 is: 0.5412 O None of…

A: Option A is correct

Q: Find the values of x in the interval [0, 2π] that satisfy: sin² x-1=0 O x = 0 and x = πT x = TT/4…

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Q: 6/155) cos 0, 1+ cos 29

A:

Q: 1) Verify the power reducing formula, cos-x= 0:1 holds true for x = and x= =.

A: Here I am attaching image so that you understand each and every step.

Q: 3 Find the exact values of s in the interval [0,2x) that satisfy the given condition cos 2s = л 11т…

A: Topic : trigonometry

Q: Determine ƒ such that (1, sin 1) is on the graph of f and e2z + 5e² + sin æ ifx 0

A: When derivative function is given, to find the actual function integrate the given derivative of the…

Q: 1. Determine the absolute extrema of f(x) = (cos x − 1) sinx on [0, π], if they exist, and identify…

A: As per our policy, we are allowed to attempt only first question. Kindly post remaining question…

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: "ind no E N such that for all n > no sin(n + n² cos(n²)) n3 – 5n4 – 20 -

A: Find n0

Q: 18 a Show that / (1 + sin 2zx)dx = n, for all positive integers n. b Sketch y = 1 + sin 27x, and…

A:

Q: Show that the functions have local extremevalues at the given values of θ, and say which kind of…

A: Consider the given function is h(θ)=3cosθ2 for the range 0≤θ≤2π. The first derivative of the…

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: if x <1 x – 1 | f(æ) = T In x -) if1<x < 4 4 In 2 %3D a cos 27rx + b sin( 2" if 4 < x | where a and…

A:

Q: - Find the extreme value/s of f(x) = 4 sin² x cos 2x on (37 3π 5π 22

A: This is a problem related to maxima and minima of the function. Based on the general formula and…

Q: 1. a. Let In = exp (nx) sin (exp (x)) dx. By making an appropriate substitution, or otherwise,…

A:

Q: 4. a. Show that for 0 Va² + sin² a. b. Generalize the result in part (a).

A:

Q: 3' (a) Recall that a' = a' · In a for any a > 0. Evaluate dt. 1 + 32t T/8 (b) Evaluate (cos 4x)(V5 +…

A: Given: (a)∫3t1+32tdt (b)∫0π8cos4x5+sin4x7dx

Q: 28- For What Values of a anel k is a X <-1 ー<くく -2 Cx) = ax-b 3 Continuons at every X ?

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Q: Q1: (1) Show that the value of € (-1, π), for which: sin² 0 1+ sin² 0 sin²0 1 + cos²0 cos² 0 cos²0 4…

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Q: Find the exact values of sin and cos for sin e= on the interval 0° s es 90°. 2 sin 0.1691, cos…

A: We know that,  cosθ=1-sin2θ         cosθ&gt;0 in 0°≤θ≤90°        ⋯⋯1sinθ2=1-cosθ2…

Q: 2. Find all values of 0 in the interval [0°, 360°) that have the given function value tan 0 = -V3.

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Q: Suppose m and n are constants and m #n. Evaluate sin(mx) cos(nx) dx-

A: To evaluate we first multiply and divide by 2 the given function.  Then use the formula: 2(sin(mx)…

Q: 2.Expand the following in powers of x i)√(1 + sinx) ii)e cosx iii) cos x²+

A: Given data:- (i); 1+sinx (ii); excosx (iii); cos-1x2-1x2+1

Q: (-1)"x2" Σ (2n)! 1. For all x E R, cos x = n=0 +00 (-16)"(4n+1) (2n)! c. Use the result in item 1(b)…

A: We have to prove the sum of series.

Q: Consider (e cos 4 + ie sin 4) (11+12:) cos 4+ ie sin 4) (11+12i) 3DZ What is the general value of z?…

A: Euler's formula is given as that e^(ix) = cos(x) + i sin(x).

Q: f (x) = cos(3r)[2 + sin(3r)]* on 0, O e*(16) - e*(7) O e*(-16) - e*(7) ) 1.52x10*(-4) O In100 - 9 O…

A: ∫avg=1b-a∫ab(cos3x)(2+sin(3x))4  ∫avg= 1π6-0∫0π/6cos3x2+sin(3x)4 dx        ....(1) Now to calculate…

Q: 1. For non-negative integers n, let k +1 k +2 Σsin n +2 n sin k = 0 n +2 f(n) = Lsin-( k +1, k = 0 n…

A: Given f(n)=∑k=0nsink+1n+2πsink+2n+2π∑k=0nsin2k+1n+2π

Q: Suppose sin(0) = and 0 < 0<T/2. Compute cos(0). (Hint: SOH-CAH-TOA) V3 3 3

A:

Q: Find the solution of the following problem accurate to within 10 using the Secant method. In(x – 1)…

A: Given function is ln(x-1)+cos(x-1)=0 and 1.3≤x≤2 So f(x)=ln(x-1)+cos(x-1) Let x0=1.3, x1=2 So…

Q: Consider the image attached of the functions f(x) = 12*cos(2x+1)-6 and g(x) = -1. Set f(x) = g(x)…

A: Given that, the functions are fx=12cos2x+1-6 and gx=-1. Equate the functions to find the solution x…

Q: 1. Determine the absolute extrema of f(z) = (sin a – 1) cos z on -. 2 if they exist, and identify…

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Q: X has a solution in Use the Intermediate Value Theorem to show that cos(x) (0, - 2

A: Using Intermediate Value Theorem to show that cos(x) =x is a solution in0,π2.

Q: a Show that (sin(m + n)x + sin(m – n)x) b Hence, without using the product rule, differentiate sin…

A: We will find out the required values.

Q: given f (x) = 3/5 cos (6x), what is the largest possible value for f (x)?

A: Given query is to find the largest possible value of the function.

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

A: First note the results derivative of log x is 1x and derivative of cos(x) is -sin(x). The proof of…

Q: Find the intervals on which f(x) = 5 x + 5 cos(r) increases for 0 < a < 2n.

A: f(x) is increasing if f'(x)&gt;0 in the interval   Note that  -1 less than equal to (sinx) less than…

Q: 7.32. Prove that for 05sr, -( . ) cos 2 cos (a) a(e -a) = sin 3r sin ba (6) (r-) - 7.33. Use Problem…

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Q: Write out Taylor's formula around 0 of degree three, f(x) = P3(x) + E3(x), for f(x) = = COs x.

A: Given that fx=cosx. To determine the Taylor's formula around 0 of degree three. That is, fx=P3x+E3x…

Q: х If f(x) = cos(1? + 1) dt, then find f'(x). %3D Select one: O A. None of the others O B. sin(x + 1)…

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Q: 8. Find all values of 0, if 0 is in the interval [0°, 360°) and cos 0 V3 2

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Q: 5) Determine the following indefinite integral cos (3x) (1 – sin(3x))(2 – sin(3x)) -dx

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Q: 2. Consider finding the root of f(x) = e*(3.2 sin(x) - 0.5 cos(x)) on the interval [3, 4], this time…

A: 2 Given function is fx=e−x3.2sinx−0.5cosx. We have to find the roots of the function fx on the…

Q: 16. So the maximum occurs at an angle of 0, but only if the angle is small enough. Explain why that…

A: Given: v=2μgscosθ+μsinθ Here μ is the coefficient of friction, g is the acceleration due to gravity…

Q: Q1\ Prove the left side equal to right side in the following : * L-1 (s² + a²y² - [sin at + at cos…

A: We need to show that:L-1s2s2+a22=12asinat+at cos atWe will use convolution theorem to show the…