Answer:
Step-by-step explanation:
_____
If you draw a triangle with a hypotenuse of 1 and an "adjacent" leg of "cos", then using the Pythagorean theorem, you can see that the "opposite" leg will be √(1-cos²) and the tangent will be (√(1-cos²))/cos. Whether or not you're allowed to draw such a triangle on paper, you can certainly do it in your mind.
Answer:
200
Step-by-step explanation:
We have:
We can rearrange the numbers to obtain:
From the left, we can factor out a negative. So:
In other words, we want to find the sum of all the odd numbers from 1 to 99.
And the sum of all the even numbers from 2 to 100.
Let's do each one individually:
Odd Terms:
We have:
We can use the arithmetic series formula, where:
Where k is the number of terms, a is the first term, and x_k is the last term.
Since it's all the odd numbers between 1 and 99, there are 50 terms.
Our first term is 1 and our last term is 99. So, the sum of all the odd terms are:
Divide the fraction. Add within the parentheses:
Multiply:
So, the sum of all the odd terms is 2500.
Even Terms:
We have:
Again, we can use the above formula.
Our first term is 2, last term is 100. And since it's from 2-100, we have 50 even terms. So:
Divide and add:
Multiply:
We originally had:
Substitute them for their respective sums:
Multiply:
Add:
Multiply:
So, the sum of our sequence is 200.
And we're done!
Note: I just found a <em>way</em> easier way to do this. We have:
Let's group every two terms together. So:
We can see that they each sum to 1:
Since there are 100 terms, we will have 50 pairs, so 50 times 1. So:
Multiply:
Pick which one you want to use! I will suggest this one though...
Edit: Typo