Sorry Its sin²
The second sum ends with +2006!, not with n
jhicks said:Recall that cos(x)=-cos(180-x). Can you perhaps express the sin^2 in another way? Remember your trig identities.
I was just speaking in general since the result is not dependent on most of the numbers in the summation. Compute 15!, 16! etc. What are the last 2 digits?
I just want to know how to calculate these sums without a calculator.
15! is definitely NOT 120. Think about this also: If you multiplied 5782389105 by 4 and looked only at the last 2 digits, is it really any different than looking at 05*4?15! is 120 and 16! 136, so it seems that 136-120 is 16. So 2006!-2005!
Use the double angle trig identity for sin^2 and you'll see what I mean.And with the Cos part, I don't really get what you mean.
I'm not English so I don't know what you mean with "Use the double angle trig identity for sin^2".jhicks said:I'm giving you hints, I'm not going to tell you how to do it. The point is to make YOU think!15! is definitely NOT 120. Think about this also: If you multiplied 5782389105 by 4 and looked only at the last 2 digits, is it really any different than looking at 05*4?Use the double angle trig identity for sin^2 and you'll see what I mean.
I'm not English so I don't know what you mean with "Use the double angle trig identity for sin^2".
Ok, so 2006! last two numbers are 00. But the problem is, I also have to add 1! up to 2005!
jhicks said:sin^2(x) = (1-cos(2x))/2. Combined with the identity cos(x) = -cos(180-x), for example, sin^2(1)+sin^2(89) = (1-cos(2))/2+(1-cos(178)/2 = (1 - cos(2) + cos(2) + 1)/2 = 1.
You will find that for much smaller factorials that this is true. 15! also has the last 2 digits of 00, for example. You can guarantee that the smallest number n such that the last 2 digits of n! are 0 that all numbers larger than n also have this property. Your task is to find the smallest n where this is true then you instantly know that all factorials larger than that do not contribute to the smallest 2 digits.
KEEP IN MIND: Any digits other than the 2 smallest in any factorial do not concern you! For example, when you want to find the last 2 digits of 7427128 multiplied by 781276, you need only consider what is 28 multiplied by 76. This is an important idea in something called modular arithmetic.
So the awnser to the sum of sin² is 45 right?
Calculating large sums without a calculator can be done by using mathematical techniques such as breaking down the sum into smaller parts, rounding numbers, and using estimation. Additionally, using trigonometric functions like sine can also help in calculating large sums.
Sine (sin) is one of the trigonometric functions that can be used to calculate angles and sides of a triangle. It can also be used to approximate values of other functions, such as logarithms or exponentials. In the context of calculating large sums, sine can help in breaking down the sum into smaller parts that are easier to calculate without a calculator.
Yes, you can use a table of values for trigonometric functions to calculate large sums without a calculator. This involves finding the corresponding values of sine for each angle in the sum and then adding them together. However, this method may not be as accurate as using other techniques such as breaking down the sum or using estimation.
Yes, some tips for calculating large sums without a calculator include breaking down the sum into smaller parts, using estimation, and using trigonometric functions like sine. It is also helpful to practice mental math techniques and to be familiar with basic mathematical operations and their properties.
Calculating large sums without a calculator can improve mental math skills and promote a deeper understanding of mathematical concepts. It can also be useful in situations where a calculator is not available or when quick estimates are needed. Additionally, it can be a fun and challenging exercise for the brain.