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Difference between revisions of "Trigonometry"

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==Basic definitions==
 
==Basic definitions==
−
Usually we call an angle <math>\displaystyle \theta</math>, read "theta", but <math> \displaystyle \theta</math> is just a variable. We could just as well call it <math> \displaystyle  a</math>.
+
Usually we call an angle <math>\theta</math>, read "theta", but <math>\theta</math> is just a variable. We could just as well call it <math> a</math>.
  
−
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.  
+
For the following definitions, the "opposite side" is the side opposite of angle <math>\theta</math>, and the "adjacent side" is the side that is part of angle <math>\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.  
+
i.e. If ABC is a right triangle with right angle C, and angle A = <math>\theta</math>, then BC is the "opposite side", AC is the "adjacent side", and AB is the hypotenuse.  
  
 
[[Image:306090triangle.gif]]
 
[[Image:306090triangle.gif]]
  
 
===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^{\circ}=\frac 12</math>.
+
The sine of an angle <math>\theta</math>, abbreviated <math>\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^{\circ} =\frac{\sqrt{3}}{2}</math>.
+
The cosine of an angle <math>\theta</math>, abbreviated <math>\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^{\circ}=\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>\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^{\circ}=2</math>. (Note that <math> \csc \theta=\frac{1}{\sin \theta}</math>.)
+
The cosecant of an angle <math>\theta</math>, abbreviated <math>\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>\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^{\circ}=\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>\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^{\circ}=\sqrt{3}</math>. (Note that <math> \cot \theta=\frac{\cos\theta}{\sin\theta}</math>.)
+
The cotangent of an angle <math>\theta</math>, abbreviated <math>\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==
−
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 <math> \displaystyle \theta </math> and the coordinates of the point we picked to be <math> \displaystyle (x,y) </math>, we have:
+
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 <math>\theta </math> and the coordinates of the point we picked to be <math>(x,y) </math>, we have:
  
−
<math> \displaystyle \sin \theta = y </math>  
+
<math>\sin \theta = y </math>  
  
−
<math> \displaystyle \cos \theta = x </math>  
+
<math>\cos \theta = x </math>  
  
−
<math> \displaystyle \tan \theta = \frac{y}{x} </math>  
+
<math>\tan \theta = \frac{y}{x} </math>  
  
−
<math> \displaystyle \csc \theta = \frac{1}{y} </math>  
+
<math>\csc \theta = \frac{1}{y} </math>  
  
−
<math> \displaystyle \sec \theta = \frac{1}{x} </math>  
+
<math>\sec \theta = \frac{1}{x} </math>  
  
−
<math> \displaystyle \cot \theta = \frac{x}{y} </math>
+
<math>\cot \theta = \frac{x}{y} </math>
  
−
Note that <math> \displaystyle (x,y) </math> is the rectangular coordinates for the point <math> (1,\theta) </math>.
+
Note that <math>(x,y) </math> is the rectangular coordinates for the point <math> (1,\theta) </math>.
  
 
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.  
 
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.  
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* [[Trigonometric substitution]]
 
* [[Trigonometric substitution]]
 
* [[Geometry]]
 
* [[Geometry]]
 +
[[Category:Trigonometry]]

Revision as of 15:49, 7 October 2007

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 $\theta$, read "theta", but $\theta$ is just a variable. We could just as well call it $a$.

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

i.e. If ABC is a right triangle with right angle C, and angle A = $\theta$, 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 $\sin \theta$, 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 $\cos \theta$, 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 $\tan \theta$, 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 $\csc \theta$, is the ratio between the hypotenuse and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, $\csc 30^{\circ}=2$. (Note that $\csc \theta=\frac{1}{\sin \theta}$.)

Secant

The secant of an angle $\theta$, abbreviated $\sec \theta$, 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 $\cot \theta$, 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 $\theta$ and the coordinates of the point we picked to be $(x,y)$, we have:

$\sin \theta = y$

$\cos \theta = x$

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

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

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

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

Note that $(x,y)$ 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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