Difference between revisions of "Trigonometric identities"

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| <math> \sin A = \frac 1{\csc A}</math> || <math> \cos A = \frac 1{\sec A}</math> || <math> \tan A = \frac 1{\cot A}</math>
 
| <math> \sin A = \frac 1{\csc A}</math> || <math> \cos A = \frac 1{\sec A}</math> || <math> \tan A = \frac 1{\cot A}</math>
 
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Another useful identity that isn't a reciprocal relation is that <math> \tan A =\frac{\sin A}{\cos A} </math>.
  
 
== Pythagorean Identities ==
 
== Pythagorean Identities ==
*<math>\displaystyle \sin^2x + \cos^2x = 1</math>
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Using the [[Pythagorean Theorem]] on our triangle above, we know that <math>\displaystyle a^2 + b^2 = c^2 </math>.  If we divide by <math> c^2 </math> we get <math> \left(\frac ac\right)^2 + \left(\frac bc\right)^2 = 1 </math> which is just <math> \sin^2 A + \cos^2 A =1 </math>.  Dividing by <math> a^2 </math> or <math> b^2 </math> instead produces two other similar identities.  The Pythagorean Identities are listed below:
*<math>\displaystyle 1 + \cot^2x = \csc^2x</math>
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*<math>\displaystyle \tan^2x + 1 = \sec^2x</math>
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{| style="height:150px; margin: 1em auto 1em auto"
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|-
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|<math>\displaystyle \sin^2x + \cos^2x = 1</math>
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|-
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|<math>\displaystyle 1 + \cot^2x = \csc^2x</math>
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|-
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|<math>\displaystyle \tan^2x + 1 = \sec^2x</math>
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|}
  
 
== Angle Addition Identities ==
 
== Angle Addition Identities ==
 
*<math>\displaystyle \sin \theta \cos \gamma + \sin \gamma \cos \theta = \sin \left(\theta+\gamma\right)</math>
 
*<math>\displaystyle \sin \theta \cos \gamma + \sin \gamma \cos \theta = \sin \left(\theta+\gamma\right)</math>
*<math>\displaystyle \cos \theta \cos \gamma - \sin theta \sin gamma = \cos \left(\theta+\gamma\right)</math>
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*<math>\displaystyle \cos \theta \cos \gamma - \sin theta \sin \gamma = \cos \left(\theta+\gamma\right)</math>
*<math>\displaystyle \frac{\tan \theta + \tan gamma}{1-\tan\theta\tan\gamma}=\tan\left(\theta+\gamma\right)</math>
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*<math>\displaystyle \frac{\tan \theta + \tan \gamma}{1-\tan\theta\tan\gamma}=\tan\left(\theta+\gamma\right)</math>
  
 
== Even-Odd Identities ==
 
== Even-Odd Identities ==

Revision as of 08:48, 24 June 2006

Trigonometric identities are used to manipulate trig equations in certain ways. Here is a list of them:

Basic Definitions

The six basic trigonometric functions can be defined using a right triangle:

Righttriangle.png

The six trig functions are sine, cosine, tangent, cosecant, secant, and cotangent. They are abbreviated by using the first three letters of their name (except for cosecant which uses $\csc$). They are defined as follows:

$\sin A = \frac ac$ $\csc A = \frac ca$
$\cos A = \frac bc$ $\sec A = \frac cb$
$\tan A = \frac ab$ $\cot A = \frac ba$

Reciprocal Relations

From the last section, it is easy to see that the following hold:

$\sin A = \frac 1{\csc A}$ $\cos A = \frac 1{\sec A}$ $\tan A = \frac 1{\cot A}$

Another useful identity that isn't a reciprocal relation is that $\tan A =\frac{\sin A}{\cos A}$.

Pythagorean Identities

Using the Pythagorean Theorem on our triangle above, we know that $\displaystyle a^2 + b^2 = c^2$. If we divide by $c^2$ we get $\left(\frac ac\right)^2 + \left(\frac bc\right)^2 = 1$ which is just $\sin^2 A + \cos^2 A =1$. Dividing by $a^2$ or $b^2$ instead produces two other similar identities. The Pythagorean Identities are listed below:

$\displaystyle \sin^2x + \cos^2x = 1$
$\displaystyle 1 + \cot^2x = \csc^2x$
$\displaystyle \tan^2x + 1 = \sec^2x$

Angle Addition Identities

  • $\displaystyle \sin \theta \cos \gamma + \sin \gamma \cos \theta = \sin \left(\theta+\gamma\right)$
  • $\displaystyle \cos \theta \cos \gamma - \sin theta \sin \gamma = \cos \left(\theta+\gamma\right)$
  • $\displaystyle \frac{\tan \theta + \tan \gamma}{1-\tan\theta\tan\gamma}=\tan\left(\theta+\gamma\right)$

Even-Odd Identities

Prosthaphaersis Indentities

(Otherwise known as sum-to-product identities)

Other Identities

  • $|1-e^{i\theta}|=2\sin\frac{\theta}{2}$

See also

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