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At it's most basic, trigonometry has to do with the properties of triangles. "Trigonos" is the same root used for the word triangle and means three sided, while "metry" is a suffix used to mean measurement - like in the word geometry. Trigonometry is the study of how the side length and angles in a triangle relate to one another. Most commonly, right triangles are used. A right triangle has one angle that is 90 degrees (the same as a corner of a square). A 90 degree angle is the same as a right angle. |

Although each of these triangles is different, they are all right triangles - the 90 degree angle is marked with a small square to show this.
Suppose you wish to know the height of a tree. Look at the following picture of the situation: You want to find h and you know only d (the distance to the tree) and A (the angle to the top of the tree. The trigonometric relationship tan (short for tangent) will tell you the answer. The tangent of angle A is equal to h/d. Algebra allows us to rearrange this to get an equation: tan(A) x d = h.
If a class of ten calculus students has their IQs measured as 120, 110, 100, 99, 132, 110, 97, 105, 115, and 106, what is the average IQ in that class? A simple average is calculated by adding up all the data points (IQs) and then dividing by the number of measurements taken. The sum of the ten IQs is 1,094 and this sum, divided by 10 equals the average, 109.4.
In the above diagram, how far is it to the top of the tree from the corner where angle A is? (This is the unlabeled line, the hypotenuse of the right triangle above.) The relationship cosine (abbreviated cos) relates d (the side of the triangle we know) to the hypotenuse, which is the distance we want to find. The relationship is given as cos(A) = d/hypotenuse. We can rearrange this so that the value we want to find is on one side of the rearranged equation: hypotenuse = d/cos(A) and get an answer directly.
No one person invented trigonometry. The techniques developed in many civilizations and demonstrate how useful trigonometry is. The more advanced versions came out of India, but in the form we use it today (as ratios of side length and angles) trigonometry is of fairly modern origin. The trigonometric functions - sine, cosine and tangent - have found use in many fields and are so practical that most calculators include keys for these functions (sin, cos, tan).
Trigonometry as it is usually taught depends on flat triangles. The relationships between side lengths and angles is only valid when a triangle is drawn on a flat plane. Drawing the same triangle on a sphere (or a donut) will give an incorrect answer. This is because, on a plane surface, the interior angles of all triangles add up to 180 degrees. On other surfaces, this isn't necessarily so. On proof that the world is nearly a sphere would be to draw a triangle with one side on the equator and the other two sides going up on opposite sides of the planet. The two lines drawn up, due north (right angles from the equator) will meet at the north pole. They will join to make a third right angle. Although it will be huge, this triangle will violate the law that all triangles have interior angles that add up to 180 degrees. In fact, this triangle's angles will add up to 270 degrees (3 x 90 degrees). The study of surfaces based on these types of properties is called topology and can tell us whether or not our 3-dimensional space is curved or flat, even though we cannot see it directly. |

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