Trigonometry

Providing that the pole is on level ground you have the outline of a right angled triangle with an adjacent side of 92 ft (the shadow of the pole) and a opposite side of 60 ft (the height of the pole). To find the angle of elevation use the tangent ratio. Tangent = Opposite/Adjacent Tangent = 60/92 = 0.652173913 Tan-1(0.652173913) = 33.11134196 degrees Therefore the angle of elevation is 33o correct to two significant figures.

A function f(x) is even if f(-x) = f(x). A graph of f(x) would be symmetric about the y-axis (vertical symmetry about x=0). f(x) need not be "well-behaved" or even continuous, unlike the examples given in Wikipedia article on "Even and odd functions". The article does make this clear - under "Some facts".

tan^2(x)

Proof:

cos^2(x)+sin^2(x)=1 (Modified Pythagorean theorem)

sin^2(x)=1-cos^2(x) (Property of subtraction)

cos^2(x)-1/cos^2(x)=?

sin^2(x)/cos^2(x)=? (Property of substitution)

sin(x)/cos(x) * sin(x)/cos(x) = tan(x) * tan(x) (Definition of tanget)

= tan^2(x)

Suppose csc(x)*sin(x) = cos(x)*cot(x) + y

then, ince csc(x) = 1/sin(x), and cot(x) = cos(x)/sin(x),

1 = cos(x)*cos(x)/sin(x) + y

so y = 1 - cos2(x)/sin(x) = 1 - [1 - sin2(x)]/sin(x) = [sin2(x) + sin(x) - 1]/sin(x)

3

No. It can be used on any triangle.

The cosine function on a right triangle is Adjacent leg divided by the hypotenuse of the triangle.

90 degrees is one

0.5735764264 (nearest 10 decimal places).

sin(t) = 2/3

sin2(t) + cos2(t) = 1

so cos(t) = ± sqrt[1 - sin2(t)]

but because t is in the first quadrant, cos(t) > 0 so cos(t) = + sqrt[1 - sin2(t)]

= sqrt[1 - 4/9] = sqrt[5/9] = sqrt(5)/3

Then sec(t) = 1/cos(t) = 1/sqrt(5)/3 = 3/sqrt(5) = 3*sqrt(5)/5

40 degrees

They do not. They are not animate nor sentient and so they are incapable of communicating.

Cosecant(115 deg) = 1/sine(115 deg) = 1/0.9063 = 1.1034

with a sine wave

A function cannot be one to many.

Suppose y = tan(x)

Now, since tan(x) = tan(x + pi)

then tan(x + pi) = y

But that means arctan(y) can be x or x+pi

In order to prevent that sort of indeterminacy, the arctan function must be restricted to an interval of width pi.

Any interval of that width would do and it could have been restricted to the first and second quadrants, or even from -pi/4 to 3*pi/4. The problem there is that in the middle of that interval the tan function becomes infinite which means that arctan would have a discontinuity in the middle of its domain. A better option, then, is to restrict it to the first and fourth quarters. Then the asymptotic values occur at the ends of the domain, which leaves the function continuous within the whole of the open interval.

If r-squared = theta then r = ±sqrt(theta)

It means the ratio of the opposite angle to the hypotenuse of a triangle for angle "x". This is for a right triangle.

-16t2 + 64t + 1224 = 0

Multiply both sides by -1

16t2 - 64t - 1224 = 0

Divide both side by 8

2t2 - 8t - 153 = 0

Cannot be factored so use the formula (-b (+ or -)(root of b2 - 4ac)) / 2a

sinθ = O/H

cosθ = A/H

tanθ = O/A

cosθ*tanθ = A/H * O/A = AO/AH = O/H

Therefore, sinθ = cosθtanθ = O/H

(Notes: O=opposite, A=adjacent, H=hypotenuse)

Cos(120) = -0.5

tan(20 deg) = 0.3640 approx.

You seem not to be aware but you could have obtained the answer much more easily and quickly by using the calculator that comes as part of your computer.

Yes, that is why they are called "principal". The domains are restricted so that the functions become injective.

Mensuration is often based on making use of a model or base object that serves as the standard for making the calculations. From that point, advanced mathematics is employed to project measurements of length, width, and weight associated with like items. The end result is data that can help to make the best use of resources available today while still planning responsibly for the future.

While mensuration is normally associated with measurements within the timber industry, the general principles can be applied in other venues as well. For example, the basics of mensuration can be used to project any type of phenomenon where growth of some sort is anticipated. Thus, mensuration may be used to project learning curves, the process of managing any type of renewable resource, or even something as simple as a projected average growth pattern for an individual.

In general, the utilization of the principles of algebra and geometry in the measuring process are capable of providing reliable data that is based on the existence of a specified set of factors. However, it is important to note that mensuration is not the only approach that is used to project future growth and volume. Because there is always the chance for unexpected elements to enter the process, the measurements obtained from the process of mensuration are normally considered a baseline. Predictions of future patterns that do factor in acts of nature and other volatile factors are then created using the results of the mensuration process as the foundation rather than the sole projection of the ultimate outcome.

2 sin^2 theta = 1/4

sin^2 theta = 1/8

sin theta = sqrt(1/8)

theta = arcsin(sqrt(1/8))