Trigonometry

A function can map each element in the domain to only one element in the codomain or range. A relation is not so restricted.

A simple non-mathematical illustration:

relation: y = biological parent(x)

function: z = biological mother(x)

Leaving aside complications from surrogacy or other exceptional situations, each person has only one natural mother. Siblings may share the same natuarl mother but they are different elements of the domain.

However, for each person, there are two biological parents. The relationship or mapping is said to be one to many, and is therefore not a function.

sin(theta) is the y-coordinate of the intersection of a line forming an angle with the positive x-axis and the unit circle (i.e., circle of radius one, centered at the origin). The unit circle has its highest y-value at the point (0,1). Drawing the unit circle on a graph makes this obvious.

Assume that there were a higher y-value than one possible. y>1 --> y2>1. Because x2>=0 for all real x, y2>1 --> x2+y2>1, which is impossible for a point on the unit circle (since for all points on the unit circle (x,y), x2+y2=1).

Among the scientific fields that make use of trigonometry are these:

acoustics, architecture, astronomy (and hence navigation, on the oceans, in aircraft, and in space; in this connection, see great circle distance), biology, cartography, chemistry, civil engineering, computer graphics, geophysics, crystallography, economics (in particular in analysis of financial markets), electrical engineering, electronics, land surveying and geodesy, many physical sciences, mechanical engineering, machining, medical imaging (CAT scans and ultrasound), meteorology, music theory, number theory (and hence cryptography), oceanography, optics, pharmacology, phonetics, probability theory, psychology, seismology, statistics, and visual perception

Congruent (APEX) :P

Oscillation is a common phenomenon in physics.Sound and electromagnetic radiation (radio, light, x-rays etc propagate as sinusoidal waves which are oscillations about a mean value. Springs, pendulums (penduli?) oscillate about their rest position in simple harmonic motion, which is oscillation about the mean.

instead of using mesh loop analysis, because most calculators don't operate in variable and complex mode at the same time, you have to use substitution.

To measure phase angle on a scope, connect the two inputs to the two sources, trigger off of one, and watch both. Measure the period of one and compare crossover difference with the other as a percentage of the period of the first, times 360 degrees.

Make sure the peak to peak offset for both inputs is equal, otherwise the crossover voltage won't be right.

Note that the period is 100% and is equivalent to 360 degrees. One way to make the calculation easier is to adjust the horizontal calibration so the first input has a period equal to 36 minor divisions - each division will equal 10 degrees in this case.

Also note, if you are comparing voltage and current, that your polarity is correct. This can be an issue if you are measuring across a small series resistor - the high side is voltage and the high to low side differential is current - but it is inverted.

There is insufficient information in the question to answer it. Please restate the question and provide more information, such as some of the sides and/or some of the other angles.

Several great mathematicians have made contributions to trigonometry. Pitiscus wrote books on plane and spherical trigonometry, and Hipparchus produced a table of chords.

The domain is the set of values of the input while the range is the set of output values.

The earth is divided into the Northern and Southern hemispheres by the equator. It is divided into the Eastern and Western hemispheres by the Greenwich meridian and the 180 degree longitude. These then form the NE, NW, SE and SW quadrants.

Optics deals with light waves, and all waves relate in some way to trigonometry. Also, the reflection and refraction of light involves trigonometry.

Calculus is made up of Trig and Algebra. Most people you ask will say that the hardest part of calculus is the algebra. The best advice I can give is to know your unit circle and Pythagoreans Theorem well.

You should really try to solve this yourself first, in order to maximize the value in doing that. If you read this solution, please understand each step before proceeding to the next step, otherwise the lesson will be lost to you.

To determine the required elevation of an observer in order that he may be able to see an object on the earth thirty miles away, assuming the earth's radius is 3956 mi and the earth is a smooth sphere, first draw the triangle involved.

This is a right triangle, where the hypotenuse is the radius of the earth plus the elevation of the observer. One side is the radius of the earth to the object. The other side is the line of sight distance from the observer to the object, which is greater than 30 miles. That line of sight is tangent to the earth's circumference, at the point of the object, so the angle of line of sight relative to the radius at the object is 90 degrees.

The angle at the center of the earth is 360 degrees times 30 miles divided by the circumference of the earth, 2 pi 3956, or 24856, which is an angle of 0.4345 degrees.

The hypotenuse of a right triangle with one angle of 0.4345 degrees and side of 3956 is 3956 divided by cosine 0.4345 degrees, or 3956.1138 miles.

Subtract the radius, 3956 miles, and you get 0.1138 miles, or 601 feet.

Not asked, but answered for completeness, is that the line of sight distance is hypotenuse times sine 0.4345 degrees, or 30.0005751 miles, or 3.04 feet more than 30 miles.

Basic arithmetic has been developed by nearly every culture, so there isn't one definite person who discovered it.

When z = 8, z lies straight along the real axis, so r = 8 and theta = 0°

Yes, it is called arcsin.

Leonhard Euler ,Gaspard Monge ,Jean-Victor Poncelet , Jakob Steiner

(the side opposite the angle) divided by (the side adjacent to the angle) = tangent of the angle

(the side opposite the angle) divided by (the hypotenuse of the triangle) = sine of the angle

(the side adjacent to the angle) divided by (the hypotenuse of the triangle) = cosine of the angle

Once you have the sine OR the cosine OR the tangent of the angle, you can get the measurement of the angle on a scientific calculator, or look it up in a table of trig functions in a book.

Check out these articles for a simple free tool and tutorial that will make trig simple enough for ANYBODY to understand!

http://www.ehow.com/how_5520340_memorize-trig-functions-losing-mind.html

http://www.ehow.com/how_5227490_pass-mind-part-unknown-sides.html

http://www.ehow.com/how_5428511_pass-part-ii-unknown-angles.html

In Epytoma…in Almagestum Ptolomei, the abridgment of Ptolemy's Almagest which was completed by his student, Regiomontanus, he replaced chords by sines, and calculated tables of sines for every minute of arc for a radius of 600,000 units. He made his observations with very simple instruments, using an ordinary plumb-line to measure the angles of elevation of the stars. He also introduced a mathematical innovation by using Hindu-Arabic numerals in his sine tables, the first transition from the duodecimal to the decimal system. Peuerbach noted several errors in Ptolemy's calculations, but remained a devotee of the ancient Greek mathematician.

Write up of geometry in real life

Pi radians is 180 degrees. So if you have theta in radians, multiply by 180/Pi

Slope is rise divided by run and percent is rise divided by run then multiply by 100 to change to % slope. As a check to your data, (3.2/40)*100 does equal 8%, so it is confirmed correct. You simply have a right triangle with height of 3.2, length of 40, and the length of the hypotenuse needs to be found to answer the question. The solution is sqrt(3.22+402) which is 40.1278 ft.

9,3,6 The dimensions given above would not be suitable for a right angled triangle which presumably the question is asking about. The dimensions suitable for a right angled triangle in the question are: 9, 12, 15.

The bearing of a point B, from a point A, is based on the angle that the straight line AB makes with the line AN which goes due North from A. The angle is measured in the clockwise direction and is normally expressed as a 3 digit number (including leading zeros).