Answer:
See Below.
Step-by-step explanation:
We want to verify the identity:
Note that the left-hand side is a perfect square trinomial pattern. Namely:
If we let <em>a</em> = csc(x) and <em>b</em> = cot(x), we can factor it as such:
Let csc(x) = 1 / sin(x) and cot(x) = cos(x) / sin(x):
Combine fractions:
Square (but do not simplify yet):
Now, we can make a substitution. Let <em>u</em> = <em>x</em> / 2. So, <em>x</em> = 2<em>u</em>. Substitute:
Recall that cos(2u) = 1 - sin²(u). Hence:
Simplify:
Recall that sin(2u) = 2sin(u)cos(u). Hence:
Square:
Cancel:
Since sin(u) / cos(u) = tan(u):
We can substitute <em>u</em> back for <em>x</em> / 2:
Hence proven.
What model is he building
Let's start by first remembering what a "geometric sequence" is. A geometric sequence is a sequence of numbers that multiply by the same number each time. For example,
2, 4, 8, 16
is a geometric sequence since each number is multiplied by 2 from one to the next, but
2, 4, 6, 8
is NOT a geometric sequence. We are adding 2 each time, but not multiplying, so it doesn't count. We have to be multiplying to have a geometric sequence.
[A] Let's look at A, are we multiplying by the same thing?
6*3 = 18
18*3 = 54
54*3 = 162
162*3 = 486
Yes! We are multiplying by 3 each time. This is a geometric sequence!
[B] Are we multiplying by the same thing?
2*3/2 = 3
3*3/2 = 9/2 which is not 5.
NOPE
[C] Are we multiplying by the same thing?
2*5/2 = 5
5*5/2 = 25/2 which is not 8
NOPE
[D] Are we multiplying by the same thing?
-4*1/2 = -2
-2*1/2 = -1
-1*1/2 = -0.5
-0.5*1/2 = -0.25
-0.25*1/2 = -0.125
Yep, we are multiplying by 1/2 each time! This is a geometric sequence!
40^2 - 32^2=576
square root of 576 will be 24 so 24 is y
(4x - 1)2 = 11
(4x - 1) = 22
4x - 1 = 22
4x = 23
x = 5.75
