Ans: We will use appropriate multiple angle formulae to show that, if cos 1?, sin 1?, or tan 1? is rational, then the cosine or tangent of some multiple of 1? must also be rational. By choosing a multiple of 1? for which a suitable trigonometric function is known to be irrational, we can derive a contradiction and thereby establish each result. cos 1?

De Moivre's theorem states that for any real number x and any integer n,

cos nx + i sin nx = (cos x + i sin x)n

Expanding the right-hand side using the binomial theorem, and equating real parts, we have

cos nx = cosnx − [n(n − 1)/2] cosn−2x sin2x + ...

Given that sin2x = 1 − cos2x, we can thereby express cos nx as a polynomial in cos x, with integer coefficients. Hence, cos x rational implies cos nx rational. Equivalently, cos nx irrational implies cos x irrational.

Taking n = 30 and x = 1?, since we know cos 30? = root 3/2 is irrational, it follows that cos 1? is irrational. sin 1?

Building on the above result, since cos 2x = 1 − 2 sin2x, we have

cos 2nx irrational implies cos 2x irrational implies sin2x irrational implies sin x irrational.

Taking n = 15 and x = 1?, it follows that sin 1? is irrational. tan 1?

The standard addition formula for tangents tan(a+b) = (tan a + tan b)/(1 - tan a tan b)

tells us that, if tan a and tan b are rational, then tan(a + b) is rational. (Of course, we must also have tan a, tan b, and tan(a + b) defined, so that tan a tan b not equal to 1.)

We know that tan 30? = 1 /root 3 is irrational. Since 30? can be built up as a series of binary sums, beginning with 1? and 1?, it follows, by contradiction, that tan 1? is irrational.

If you have the better answer, then send it to us. We will display your answer after the approval.
Rules to Post Answers in CoolInterview.com:-

There should not be any Spelling Mistakes.

There should not be any Gramatical Errors.

Answers must not contain any bad words.

Answers should not be the repeat of same answer, already approved.

Q.An urn contains a number of colored balls, with equal numbers of each color. Adding 20 balls of a new color to the urn would not change the probability of drawing (without replacement) two balls of the same color.

How many balls are in the urn? (Before the extra balls are added.)

Q.Consecutive fifth powers (or, indeed, any powers) of positive integers are always relatively prime. That is, for all n > 0, n5 and (n + 1)5 are relatively prime. Are n5 + 5 and (n + 1)5 + 5 always relatively prime? If not, for what values of n do they have a common factor, and what is that factor?

Q.An absentminded professor buys two boxes of matches and puts them in his pocket. Every time he needs a match, he selects at random (with equal probability) from one or other of the boxes. One day the professor opens a matchbox and finds that it is empty. (He must have absentmindedly put the empty box back in his pocket when he took the last match from it.) If each box originally contained n matches, what is the probability that the other box currently contains k matches? (Where 0 less than or equal to k less than or equal to n.)

Q.Let x be a real number and n be a positive integer. Show that [x] + [x + 1/n] + ... + [x + (n−1)/n] = [nx], where [x] is the greatest integer less than or equal to x.

Q.Two perfect logicians, S and P, are told that integers x and y have been chosen such that 1 < x < y and x+y < 100. S is given the value x+y and P is given the value xy. They then have the following conversation.

P: I cannot determine the two numbers. S: I knew that. P: Now I can determine them. S: So can I.

Given that the above statements are true, what are the two numbers? (Computer assistance allowed.)

Q.A confused bank teller transposed the dollars and cents when he cashed a check for Ms Smith, giving her dollars instead of cents and cents instead of dollars. After buying a newspaper for 50 cents, Ms Smith noticed that she had left exactly three times as much as the original check. What was the amount of the check? (Note: 1 dollar = 100 cents.)

Q.A 12 by 25 by 36 cm cereal box is lying on the floor on one of its 25 by 36 cm faces. An ant, located at one of the bottom corners of the box, must crawl along the outside of the box to reach the opposite bottom corner. What is the length of the shortest such path?

Note: The ant can walk on any of the five faces of the box, except for the bottom face, which is flush in contact with the floor. It can crawl along any of the edges. It cannot crawl under the box.

Q.Let p(x) be a polynomial with integer coefficients. Show that, if the constant term is odd, and the sum of all the coefficients is odd, then p has no integer roots. (That is, if p(x) = a0 + a1x + ... + anxn, a0 is odd, and a0 + a1 + ... + an is odd, then there is no integer k such that p(k) = 0.)

Q.Suppose xy = yx, where x and y are positive real numbers, with x < y. Show that x = 2, y = 4 is the only integer solution. Are there further rational solutions? (That is, with x and y rational.) For what values of x do real solutions exist?

Q.Find the smallest natural number greater than 1 billion (109) that has exactly 1000 positive divisors. (The term divisor includes 1 and the number itself. So, for example, 9 has three positive divisors.)

Q.Given any sequence of n integers, show that there exists a consecutive subsequence the sum of whose elements is a multiple of n. For example, in sequence {1,5,1,2} a consecutive subsequence with this property is the last three elements; in {1,−3,−7} it is simply the second element.