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Featured 5 months ago

The given function representing the height

(a) h is a cosine function of t, So it will have maximum value when

and the maximum height of the swing becomes

(b) it takes

c) The minimum height of the swing will be achieved when

Minimum height

(d). Again

Here

e) For

when

So in

f) The height of the swing at 10 s can be had by inserting

Featured 5 months ago

This creates a 45-45-90 triangle, also known as a *right isosceles triangle*. This is a very special triangle, and we know that both of its legs will be the same length, and the hypotenuse will be the length of one of the legs times

To make things easier for us, we will say the legs are both 1, and the hypotenuse, therefore, is

Now that we have a triangle with our 3 sides labeled (and our angle in red), we can find all 6 trig functions:

#sin(pi/4) = "opposite"/"hypotenuse" = 1/sqrt2 = sqrt2/2#

#cos(pi/4) = "adjacent"/"hypotenuse" = 1/sqrt2 = sqrt2/2#

#tan(pi/4) = "opposite"/"adjacent" = 1/1 = 1#

Now, the other three functions are just the inverses of the first three. Cosecant is the inverse of sine, secant is the inverse of cosine, and cotangent is the inverse of tangent. To see what I mean, look at this:

#csc(pi/4) = "hypotenuse"/"opposite" = sqrt2/1 = sqrt2#

#sec(pi/4) = "hypotenuse"/"adjacent" = sqrt2/1 = sqrt2#

#cot(pi/4) = "adjacent"/"opposite" = 1/1 = 1#

So there you have it! All 6 trig functions can be evaluated this way just by drawing a triangle and knowing its side lengths.

*Final Answer*

Featured 4 months ago

A few thoughts...

It was known and studied by Euclid (approx 3rd or 4th century BCE), basically for many geometric properties...

It has many interesting properties, of which here are a few...

The Fibonacci sequence can be defined recursively as:

#F_0 = 0#

#F_1 = 1#

#F_(n+2) = F_n + F_(n+1)#

It starts:

#0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,...#

The ratio between successive terms tends to

#lim_(n->oo) F_(n+1)/F_n = phi#

In fact the general term of the Fibonacci sequence is given by the formula:

#F_n = (phi^n - (-phi)^(-n))/sqrt(5)#

A rectangle with sides in ratio

This is related to both the limiting ratio of the Fibonacci sequence and the fact that:

#phi = [1;bar(1)] = 1+1/(1+1/(1+1/(1+1/(1+1/(1+...)))))#

which is the most slowly converging standard continued fraction.

If you place three golden rectangles symmetrically perpendicular to one another in three dimensional space, then the twelve corners form the vertices of a regular icosahedron. Hence we can calculate the surface area and volume of a regular icosahedron of given radius. See https://socratic.org/s/aFZyTQfn

An isosceles triangle with sides in ratio

Featured 3 months ago

Featured 2 months ago

graph{3sin(2(x-1)) [-10, 10, -5, 5]}

If we consider

graph{3sinx [-10, 10, -5, 5]}

We will look at C next, this is the movement of the graph left or right, where a negative C value moves the graph to the right. So we move the whole graph 1 to the right in this case.

graph{3sin(x-1) [-10, 10, -5, 5]}

Finally B is stretching the graph parallel to the x axis by a factor of

So in your case B = 2, so

Then graphing this:

graph{3sin(2(x-1)) [-10, 10, -5, 5]}

Featured 1 month ago

Graph

We know that

I would assume you know how to graph a

Now, you need to graph

Imagine you have a function

What this means is that for any point

That is what this

This means that all points on *earlier* than *left* by 1 unit to obtain

To generalize:

If **right** to get

**Now, we can apply it to this question:**

We have **left** by

The blue curve is your

The red curve is your

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