Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Trigonometry"

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.)
Line 4: Line 4:
 
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 base and the [[hypotenuse]] of a triangle with the uppermost angle equal to [[theta]]. 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=\frac 12</math>.
  
 
===[[Cosine]]===
 
===[[Cosine]]===
−
The cosine of an angle <math>\theta</math>, abbreviated <math>\displaystyle \cos \theta</math>, is the ratio between the [[altitude]] and the [[hypotenuse]] of a triangle with the uppermost angle equal to [[theta]]. 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=\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 base and altitude of a triangle with the uppermost angle equal to [[theta]]. For instance, in the 30-60-90 triangle above, <math>\tan 30=\frac{1}{\sqrt{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=\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.

Basic definitions

Usually we call an angle $\displaystyle \theta$, 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 $\displaystyle \theta$ 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.

image of a 30-60-90 triangle

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, $\sin 30=\frac 12$.

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, $\cos 30=\frac{\sqrt{3}}{2}$.

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, $\tan 30=\frac{\sqrt{3}}{3}$. (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, $\displaystyle \csc 30=2$. (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, $\sec 30=\frac{2\sqrt{3}}{3}$. (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, $\cot 30=\sqrt{3}$. (Note that $\cot \theta=\frac{\cos\theta}{\sin\theta}$.)

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Trigonometry&oldid=4401"