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Difference between revisions of "User:Temperal/The Problem Solver's Resource1"

(→‎Other Formulas)
(additional details)
Line 26: Line 26:
 
<math>\tan (90-A)=\cot A</math>
 
<math>\tan (90-A)=\cot A</math>
  
 +
<math>\sec{90-A}=\csc A</math>
 +
 +
<math>\cos (90-A) = \sin A</math>
 +
 +
<math>\cot (90-A)=\tan A</math>
 +
 +
<math>\csc (90-A)=\sec A</math>
 +
 +
<math>\sin^2 A+\cos^2 A=1</math>
 +
 +
<math>\sec^2 A-\tan^2 A=1</math>
 +
 +
<math>\csc^2 A-\cot^2 A=1</math>
 +
 +
<math>\tan A=\frac{\sin A}{\cos A}</math>
 +
 +
<math>\sin^2 \frac{A}{2}=\frac{1}{2}(1-\cos A)</math>
 +
 +
<math>\cos^2 \frac{A}{2}=\frac{1}{2}(1+\cos A)</math>
 +
 +
<math>\tan \frac{A}{2}=\frac{1-\cos A}{\sin A}=\frac{\sin A}{1+\cos A}</math>
 +
 +
===Terminology===
 +
<math>\cot A=\frac{1}{\tan A}</math>, but <math>\cot A\ne\tan^{-1} A}</math>.
 +
 +
<math>\csc A=\frac{1}{\sin A}</math>, but <math>\csc A\ne\sin^{-1} A}</math>.
 +
 +
<math>\sec A=\frac{1}{\sin A}</math>, but <math>\sec A\ne\cos^{-1} A}</math>.
 +
 +
Also:
 +
 +
<math>\tan^{-1} A=\atan A=\arctan A</math>
 +
 +
<math>\tan^{-1} A=\asin A=\arcsin A</math>
 +
 +
<math>\tan^{-1} A=\asin A=\arcsin A</math>
 
===Sum of Angle Formulas===
 
===Sum of Angle Formulas===
 
<math>\sin (A \pm B)=\sin A \cos B \pm \cos A \sin B</math>
 
<math>\sin (A \pm B)=\sin A \cos B \pm \cos A \sin B</math>
Line 54: Line 90:
  
 
===Other Formulas===
 
===Other Formulas===
−
 
+
====Law of Cosines==
 
In a triangle with sides <math>a</math>, <math>b</math>, and <math>c</math> opposite angles <math>A</math>, <math>B</math>, and <math>C</math>, respectively,
 
In a triangle with sides <math>a</math>, <math>b</math>, and <math>c</math> opposite angles <math>A</math>, <math>B</math>, and <math>C</math>, respectively,
  
 
<math>c^2=a^2+b^2-2bc\cos A</math>
 
<math>c^2=a^2+b^2-2bc\cos A</math>
  
−
and
+
and:
 +
====Law of Sines====
  
 
<math>\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R</math>
 
<math>\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R</math>
  
 +
====Law of Tangents===
 +
 +
For any <math>a</math> and <math>b</math> such that <math>\tan a,\tan b \subset \mathbb{R}</math>,
 +
<math>\frac{a-b}{a+b}=\frac{\tan(a-b)}{\tan(a+b)}</math>
 +
 +
====Area of a Triangle====
 
The [[area]] of a triangle can be found by  
 
The [[area]] of a triangle can be found by  
  

Revision as of 16:19, 9 October 2007



The Problem Solver's Resource
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 1.

Trigonometric Formulas

Note that all measurements are in degrees, not radians.

Basic Facts

$\sin (-A)=-\sin A$

$\cos (-A)=\cos A$

$\tan (-A)=-\tan A$

$\sin (180-A) = \sin A$

$\cos (180-A) = -\cos A$

$\cos (360-A) = \cos A$

$\tan (180+A) = \tan A$

$\cos (90-A)=\sin A$

$\tan (90-A)=\cot A$

$\sec{90-A}=\csc A$

$\cos (90-A) = \sin A$

$\cot (90-A)=\tan A$

$\csc (90-A)=\sec A$

$\sin^2 A+\cos^2 A=1$

$\sec^2 A-\tan^2 A=1$

$\csc^2 A-\cot^2 A=1$

$\tan A=\frac{\sin A}{\cos A}$

$\sin^2 \frac{A}{2}=\frac{1}{2}(1-\cos A)$

$\cos^2 \frac{A}{2}=\frac{1}{2}(1+\cos A)$

$\tan \frac{A}{2}=\frac{1-\cos A}{\sin A}=\frac{\sin A}{1+\cos A}$

Terminology

$\cot A=\frac{1}{\tan A}$, but $\cot A\ne\tan^{-1} A}$ (Error compiling LaTeX. Unknown error_msg).

$\csc A=\frac{1}{\sin A}$, but $\csc A\ne\sin^{-1} A}$ (Error compiling LaTeX. Unknown error_msg).

$\sec A=\frac{1}{\sin A}$, but $\sec A\ne\cos^{-1} A}$ (Error compiling LaTeX. Unknown error_msg).

Also:

$\tan^{-1} A=\atan A=\arctan A$ (Error compiling LaTeX. Unknown error_msg)

$\tan^{-1} A=\asin A=\arcsin A$ (Error compiling LaTeX. Unknown error_msg)

$\tan^{-1} A=\asin A=\arcsin A$ (Error compiling LaTeX. Unknown error_msg)

Sum of Angle Formulas

$\sin (A \pm B)=\sin A \cos B \pm \cos A \sin B$

$\cos (A \pm B)=\cos A \cos B \mp \sin A \sin B$

$\tan (A \pm B)=\frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

$\sin2A=2\sin A \cos A$

$\cos2A=\cos^2 A - \sin^2 A$ or $\cos2A=2\cos^2 A -1$ or $\cos2A=1- 2 \sin^2 A$

$\tan2A=\frac{2\tan A}{1-\tan^2 A}$

Pythagorean identities

$\sin^2 A+\cos^2 A=1$

$1 + \tan^2 A = \sec^2 A$

$1 + \cot^2 A = \csc^2 A$

for all $A$.

Other Formulas

==Law of Cosines

In a triangle with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$, respectively,

$c^2=a^2+b^2-2bc\cos A$

and:

Law of Sines

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$

=Law of Tangents

For any $a$ and $b$ such that $\tan a,\tan b \subset \mathbb{R}$, $\frac{a-b}{a+b}=\frac{\tan(a-b)}{\tan(a+b)}$

Area of a Triangle

The area of a triangle can be found by

$\frac 12ab\sin C$

Back to intro | Continue to page 2



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