MIT

🔍

question:Compute cos 240^circ.

answer:First, we note that 240^circ is in the third quadrant, where cosine is negative. To find the cosine of 240^circ, we can use the fact that the cosine function has a period of 360^circ. So, we can subtract 360^circ from 240^circ to get an angle in the same cosine value in the first quadrant. 240^circ - 360^circ = -120^circ. The cosine of -120^circ is the same as the cosine of 120^circ, since cosine is an even function. We know the cosine of 120^circ is -1/2. Therefore, cos 240^circ = boxed{-frac{1}{2}}.The answer is: -frac{1}{2}

question:Yvonne and Janna were writing their 1000-word pair research paper. Yvonne was able to write x words while Janna wrote 150 more words than Yvonne. When they edited their paper, they removed 20 words and added twice as many words as they removed. They should add 30 more words to reach the research paper requirement. What is the value of unknown variable x? What is the value of unknown variable x?

answer:To solve this problem, we need to determine the value of x, which represents the number of words Yvonne wrote. Let's break down the information given: Number of words Yvonne wrote: x Number of words Janna wrote: x + 150 Number of words removed during editing: 20 Number of words added during editing: 2 * 20 = 40 Number of words needed to reach the research paper requirement: 30 Total number of words after editing: (x + x + 150) - 20 + 40 + 30 We can set up the equation as follows: Total number of words after editing = 1000 (x + x + 150) - 20 + 40 + 30 = 1000 Let's simplify and solve for x: 2x + 150 + 50 = 1000 2x + 200 = 1000 To isolate x, we subtract 200 from both sides of the equation: 2x + 200 - 200 = 1000 - 200 2x = 800 To solve for x, we divide both sides of the equation by 2: x = 800 / 2 x = 400 The value of x is 400. 400 The answer is: 400

question:What is the smallest prime number with two digits such that reversing the digits results in a composite number?

answer:The smallest two-digit prime number is 11, but reversing the digits gives another prime number, 11. The next prime number is 13, and reversing the digits gives 31, which is also prime. The next prime number is 17, and reversing the digits gives 71, which is also prime. The next prime number is 19, and reversing the digits gives 91, which is not prime. Therefore, the smallest prime number with two digits such that reversing the digits results in a composite number is boxed{19}. The answer is: 19

question:f the remainder of dividing n by 3 is 2, what is the remainder when 5n is divided by 3?

answer:If the remainder of dividing n by 3 is 2, then we can write n as 3k + 2 for some integer k. Now, let's consider 5n. We have 5n = 5(3k + 2) = 15k + 10. When we divide 15k + 10 by 3, we get a remainder of 1. Therefore, the remainder when 5n is divided by 3 is boxed{1}.The answer is: 1