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Difference between revisions of "Trigonometry"

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===Sine===
 
===Sine===
−
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>.
+
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^{\circ}=\frac 12</math>.
  
 
===Cosine===
 
===Cosine===
−
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>.
+
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^{\circ} =\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 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>.)
+
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^{\circ}=\frac{\sqrt{3}}{3}</math>. (Note that <math> \tan \theta=\frac{\sin\theta}{\cos\theta}</math>.)
  
 
===Cosecant===
 
===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>.)
+
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^{\circ}=2</math>. (Note that <math> \csc \theta=\frac{1}{\sin \theta}</math>.)
  
 
===Secant===
 
===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>.)
+
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^{\circ}=\frac{2\sqrt{3}}{3}</math>. (Note that <math> \sec \theta=\frac{1}{\cos \theta}</math>.)
  
  
 
===Cotangent===
 
===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>.)
+
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^{\circ}=\sqrt{3}</math>. (Note that <math> \cot \theta=\frac{\cos\theta}{\sin\theta}</math>.)
  
 
==Trigonometery Definitions for non-acute angles==
 
==Trigonometery Definitions for non-acute angles==

Revision as of 11:22, 8 September 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.

Basic definitions

Usually we call an angle $\displaystyle \theta$, read "theta", but $\displaystyle \theta$ is just a variable. We could just as well call it $\displaystyle a$.

For the following definitions, the "opposite side" is the side opposite of angle $\displaystyle \theta$, and the "adjacent side" is the side that is part of angle $\displaystyle \theta$, but is not the hypotenuse.

i.e. If ABC is a right triangle with right angle C, and angle A = $\displaystyle \theta$, then BC is the "opposite side", AC is the "adjacent side", and AB is the hypotenuse.

306090triangle.gif

Sine

The sine of an angle $\theta$, abbreviated $\displaystyle \sin \theta$, is the ratio between the opposite side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, $\sin 30^{\circ}=\frac 12$.

Cosine

The cosine of an angle $\theta$, abbreviated $\displaystyle \cos \theta$, is the ratio between the adjacent side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, $\cos 30^{\circ} =\frac{\sqrt{3}}{2}$.

Tangent

The tangent of an angle $\theta$, abbreviated $\displaystyle \tan \theta$, is the ratio between the opposite side and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, $\tan 30^{\circ}=\frac{\sqrt{3}}{3}$. (Note that $\tan \theta=\frac{\sin\theta}{\cos\theta}$.)

Cosecant

The cosecant of an angle $\theta$, abbreviated $\displaystyle \csc \theta$, is the ratio between the hypotenuse and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, $\displaystyle \csc 30^{\circ}=2$. (Note that $\csc \theta=\frac{1}{\sin \theta}$.)

Secant

The secant of an angle $\theta$, abbreviated $\displaystyle \sec \theta$, is the ratio between the hypotenuse and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, $\sec 30^{\circ}=\frac{2\sqrt{3}}{3}$. (Note that $\sec \theta=\frac{1}{\cos \theta}$.)


Cotangent

The cotangent of an angle $\theta$, abbreviated $\displaystyle \cot \theta$, is the ratio between the adjacent side and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, $\cot 30^{\circ}=\sqrt{3}$. (Note that $\cot \theta=\frac{\cos\theta}{\sin\theta}$.)

Trigonometery Definitions for non-acute angles

Consider a unit circle that is centered at the origin. By picking a point on the circle, and dropping a perpendicular line to the x-axis, a right triangle is formed with a hypotenuse 1 unit long. Letting the angle at the origin be $\displaystyle \theta$ and the coordinates of the point we picked to be $\displaystyle (x,y)$, we have:

$\displaystyle \sin \theta = y$

$\displaystyle \cos \theta = x$

$\displaystyle \tan \theta = \frac{y}{x}$

$\displaystyle \csc \theta = \frac{1}{y}$

$\displaystyle \sec \theta = \frac{1}{x}$

$\displaystyle \cot \theta = \frac{x}{y}$

Note that $\displaystyle (x,y)$ is the rectangular coordinates for the point $(1,\theta)$.

This is true for all angles, even negative angles and angles greater than 360 degrees. Due to the way trig ratios are defined for non-acute angles, the value of a trig ratio could be positive or negative, or even 0.

See also

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