Trigonometry
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By definition, of course, one angle of any right triangle is itself a right angle. So, if one other angle is known, then all three are known ([[I.32|The angles of a triangle sum to a straight line.]]). Every right triangle containing that one known angle is [[similar]] to every other. So, given that angle and any one side, the other two sides may be known as well. | By definition, of course, one angle of any right triangle is itself a right angle. So, if one other angle is known, then all three are known ([[I.32|The angles of a triangle sum to a straight line.]]). Every right triangle containing that one known angle is [[similar]] to every other. So, given that angle and any one side, the other two sides may be known as well. | ||
| - | [[I.47|Pythagoras]] is the key to trig. Given any two sides of a right triangle, the third can be found. We proceed in specific cases to derive values for trig functions. | + | [[I.47|Pythagoras]] is the key to trig. Given any two sides of a right triangle, the third can be found. We proceed in specific cases to derive values for trig functions. |
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| + | [[Image:Trig-triangle.png|400px|left]] | ||
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| + | [[Image:Sin-01.png]] | ||
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| + | [[Image:Cos-01.png]] | ||
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| + | [[Image:Tan-01.png]] | ||
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== Analytic Definitions == | == Analytic Definitions == | ||
Revision as of 08:19, 6 September 2007
Literally, Trigonometry is that branch of Geometry dealing with triangle measurement. There are actually many ways to measure triangles; "trig" consists of a set of special relationships among angles and sides of right triangles. These permit the rapid calculation of unknown dimensions. While the topic appears limited, trig crops up unexpectedly in so many areas of diverse disciplines that study is considered essential.
As with most geometric topics, trig can be explored traditionally, using synthetic methods; or analytically, in which direction most modern texts turn. You may wish to explore in both directions.
Basic Facts
In a sense, there is only one trigonometric relationship. Each trig function relates an angle to some ratio of lengths or distances; when one is known, all are known. For convenience, though, several functions are defined. Each function may be derived entirely from any other.
Trig functions are unary operators, as is the absolute value; as opposed to more familiar binary operators such as addition, multiplication, and exponentiation. Given an angle, each trig function produces a ratio; given a ratio, the inverse function produces an angle.
The most common trig functions are:
| Name | Notation | Name | Notation |
|---|---|---|---|
| Sine | sin θ | Cotangent | cot θ |
| Cosine | cos θ | Secant | sec θ |
| Tangent | tan θ | Cosecant | csc θ |
The first three -- sine, cosine, and tangent -- are the most common and must be memorized.
Synthetic Definitions
By definition, of course, one angle of any right triangle is itself a right angle. So, if one other angle is known, then all three are known (The angles of a triangle sum to a straight line.). Every right triangle containing that one known angle is similar to every other. So, given that angle and any one side, the other two sides may be known as well.
Pythagoras is the key to trig. Given any two sides of a right triangle, the third can be found. We proceed in specific cases to derive values for trig functions.
