Difference between revisions of "Trigonometry"
ComplexZeta (talk | contribs) m (→See also: typo) |
(→Basic definitions: Added csc, sec, and cot. Changed "base" to "opposite side" and "altitude" to "adjacent side" since the angle theta is not always the topmost angle.) |
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Usually we call an angle <math>\displaystyle \theta</math>, read "theta", but <math>\theta</math> is just a variable. We could just as well call it <math>a</math>. | Usually we call an angle <math>\displaystyle \theta</math>, read "theta", but <math>\theta</math> is just a variable. We could just as well call it <math>a</math>. | ||
− | ''image'' | + | For the following definitions, the "opposite side" is the side opposite of angle <math>\displaystyle \theta</math> and the "adjacent side" is the side that is part of angle <math>\displaystyle \theta</math> but is not the hypotenuse. |
+ | |||
+ | i.e. If ABC is a right triangle with right angle C, and angle A = <math>\displaystyle \theta</math>, then BC is the "opposite side", AC is the "adjacent side", and AB is the hypotenuse. | ||
+ | |||
+ | ''image of a 30-60-90 triangle'' | ||
===[[Sine]]=== | ===[[Sine]]=== | ||
− | The sine of an angle <math>\theta</math>, abbreviated <math>\displaystyle \sin \theta</math>, is the ratio between the | + | The sine of an angle <math>\theta</math>, abbreviated <math>\displaystyle \sin \theta</math>, is the ratio between the opposite side and the [[hypotenuse]] of a triangle. For instance, in the 30-60-90 triangle above, <math>\sin 30=\frac 12</math>. |
===[[Cosine]]=== | ===[[Cosine]]=== | ||
− | The cosine of an angle <math>\theta</math>, abbreviated <math>\displaystyle \cos \theta</math>, is the ratio between the | + | The cosine of an angle <math>\theta</math>, abbreviated <math>\displaystyle \cos \theta</math>, is the ratio between the adjacent side and the [[hypotenuse]] of a triangle. For instance, in the 30-60-90 triangle above, <math>\cos 30=\frac{\sqrt{3}}{2}</math>. |
===[[Tangent]]=== | ===[[Tangent]]=== | ||
− | The tangent of an angle <math>\theta</math>, abbreviated <math>\displaystyle \tan \theta</math>, is the ratio between the | + | The tangent of an angle <math>\theta</math>, abbreviated <math>\displaystyle \tan \theta</math>, is the ratio between the opposite side and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, <math>\tan 30=\frac{\sqrt{3}}{3}</math>. (Note that <math> \tan \theta=\frac{\sin\theta}{\cos\theta}</math>.) |
+ | |||
+ | ===[[Cosecant]]=== | ||
+ | The cosecant of an angle <math>\theta</math>, abbreviated <math>\displaystyle \csc \theta</math>, is the ratio between the [[hypotenuse]] and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, <math>\displaystyle \csc 30=2</math>. (Note that <math> \csc \theta=\frac{1}{\sin \theta}</math>.) | ||
+ | |||
+ | ===[[Secant]]=== | ||
+ | The secant of an angle <math>\theta</math>, abbreviated <math>\displaystyle \sec \theta</math>, is the ratio between the [[hypotenuse]] and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, <math>\sec 30=\frac{2\sqrt{3}}{3}</math>. (Note that <math> \sec \theta=\frac{1}{\cos \theta}</math>.) | ||
+ | |||
+ | |||
+ | ===[[Cotangent]]=== | ||
+ | The cotangent of an angle <math>\theta</math>, abbreviated <math>\displaystyle \cot \theta</math>, is the ratio between the adjacent side and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, <math>\cot 30=\sqrt{3}</math>. (Note that <math> \cot \theta=\frac{\cos\theta}{\sin\theta}</math>.) | ||
==See also== | ==See also== |
Revision as of 22:06, 23 June 2006
Trigonometry seeks to find the lengths of a triangle's sides, given 2 angles and a side. Trigonometry is closely related to analytic geometry.
Contents
Basic definitions
Usually we call an angle , read "theta", but is just a variable. We could just as well call it .
For the following definitions, the "opposite side" is the side opposite of angle and the "adjacent side" is the side that is part of angle but is not the hypotenuse.
i.e. If ABC is a right triangle with right angle C, and angle A = , then BC is the "opposite side", AC is the "adjacent side", and AB is the hypotenuse.
image of a 30-60-90 triangle
Sine
The sine of an angle , abbreviated , is the ratio between the opposite side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, .
Cosine
The cosine of an angle , abbreviated , is the ratio between the adjacent side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, .
Tangent
The tangent of an angle , abbreviated , is the ratio between the opposite side and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, . (Note that .)
Cosecant
The cosecant of an angle , abbreviated , is the ratio between the hypotenuse and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, . (Note that .)
Secant
The secant of an angle , abbreviated , is the ratio between the hypotenuse and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, . (Note that .)
Cotangent
The cotangent of an angle , abbreviated , is the ratio between the adjacent side and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, . (Note that .)