(a) Prove that (cos i sin )2 cos 2 i sin 2, where i C and i2 1 (b) Using induction, prove that for all n Z , (cos i sin )n cos n i sin n (This result is known as DeMoivre 's Theorem ) (c) Verify that 1 i 2(cos 45 i sin 45), and compute (l i)100

Question: (a) Prove that (cos + i sin )2 = cos 2 + i sin 2, where i C and i2 = -1. (b)

(a) Prove that (cos θ + i sin θ)2 = cos 2θ + i sin 2θ, where i ∈ C and i2 = -1.
(b) Using induction, prove that for all n ∈ Z+,
(cos θ + i sin θ)n = cos nθ + i sin nθ.
(This result is known as DeMoivre's Theorem.)
(c) Verify that 1 + i = √2(cos 45° + i sin 45°), and compute (l + i)100.

Step by Step Solution

★★★★★

3.39 Rating (177 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

a cos sin 2 cos 2 2isincos sin 2 cos 2 sin 2 t2sincos cos 2 a sin 2 b Sn cos sin n cos n i sin n S1 ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

954-M-L-A-L-S (7582).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Generalizing Example 2.17(c), by a trigonometric polynomial of degree in the powers of the sine and cosine functions up to degree n. degree n. (a) Use formula (3.86) to prove that any trigonometric...

(a) Let A be a 2 x 2 matrix such that tr A = 0 and = det A > 0. Prove that eA = (cos ) I + sin / A. Use Exercise 8.2.51. (b) Establish a similar formula when det A

a. Differentiate both sides of the identity cos 2t = cos 2 t - sin 2 t to prove that sin 2t = 2 sin t cos t. b. Verify that you obtain the same identity for sin 2t as in part (a) if you differentiate...

Prove the given identity. tan x + cot x = sec x cSC X Choose the correct proof below. O A. tan x + ct x sin x COS X sin 2 COS X + sin x x+ cos 2 x sin x cos x 1 sin x cos x = sec xcsC X OC. tan x+...

Prove the given identity. 1 sec 0 = tan sin e sec 0 Choose the correct proof below. 1 1 cos e O A. sec 0 sin 20 cos = sec 0 = tan sin e cos e 1 - cos 2 cos 1 1 1 - cos B. sec 0- cos 20 cos O = cos 0=...

Name class date Precalculus Unit 4 Practice Lesson 21-1 5. Make sense of problems. Suppose you know that cos u 5 20.25. 1. Make use of structure. Express each trigonometric function as a reciprocal...

25 . 0 and a are each acute angles in standard position. sin 0 = and cos a = 12 vilw 13' a) Determine cos(a + 0). b) Determine sin( 0- a). c) Explain why the direct angle measurements are not needed...

Hello can someone please help me with those question Find the exact value of the expression. 1 tan arccos = I] 1 2 tan [arccos [E] ] (Type an exact answer, using radicals as needed Verify that the...

le Edit View History Bookmarks Window Help deanza.instructure.com C Problem.. LART. "/ Word... LT Langu... My Ho.. c Cenga... Googl.. G maus... $4 (3.4, 3.5, 3.6)-2.pdf & Download @ Info lover...

I need to see how 28 36 and 47 are done please. 35. (a) then so is A + B. 17. Prove Theorem 3.2(a)-(d). 18. Prove Theorem 3.2(e)-(h). (b) Prove that if A is a symmetric non matrix, the is KA for any...

Explain the four types of selling premises.

A test rocket is launched by accelerating it along a 200.0-m incline at 1.25 m/s2 starting from rest at point A (Figure 3.45.) The incline rises at 35.0o above the horizontal, and at the instant the...

E9.1 (LO 1, 2) (Acquisition Costs of Realty) The expenditures and receipts below are related to land, land improvements, and buildings acquired for use in a business. The receipts are enclosed in...

An article in the Wall Street Journal noted that Google was planning on entering the market to provide wireless data services. According to the article, Google has said it isnt looking to supplant...

Find the indicated term of the geometric sequence. a. 2, 4, 8, 16, c; the 7th term b. 2, -10, 50, -250, c; the 9th term c. 2, 223, 6, c; the 9th term

Find the nth, or general, term. a. 1, 3, 9,...... b. 25, 5, 1,....... c. 1, -1, 1, -1, .....

Determine whether the statement is true or false. a. The sequence 2, -2 2, 4, -42, 8,....... is geometric. b. The sequence with general term 3n is geometric. c. The sequence with general term 2n is...

You've invested $2,800 in Stock Arbuckle and $3,800 invested in Stock Roscoe. The expected returns on these stocks are 8 percent and 11 percent, respectively. Calculate the expected return on the...

The balance sheet and income statement shown below are for Sneaker Inc. Note that the firm has no amortization charges, it does not lease any assets, none of its debt must be retired during the next...

Present value ( in $ ) of an annity due \ table [ [ \ table [ [ Annuity ] , [ Payment ] ] , \ table [ [ Payment ] , [ Frequency ] ] , \ table [ [ Time ] , [ Period ( years ) ] ] , \ table [ [ Nominal...