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Difference between revisions of "Trigonometric identities"

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== Double/Half Angle Identities ==
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== Double Angle Identities ==
 
Double angle identities are easily derived from the angle addition formulas by just letting <math> \alpha = \beta </math>.  Doing so yields:
 
Double angle identities are easily derived from the angle addition formulas by just letting <math> \alpha = \beta </math>.  Doing so yields:
  
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| <math> \tan 2\alpha </math> || = || <math>\frac{2\tan \alpha}{1-\tan^2\alpha} </math>
 
| <math> \tan 2\alpha </math> || = || <math>\frac{2\tan \alpha}{1-\tan^2\alpha} </math>
 
|}
 
|}
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== Half Angle Identities ==
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Using the double angle identities, we can now derive half angle identities.  The double angle formula for cosine tells us <math> \cos 2\alpha = 2\cos^2 \alpha - 1 </math>.  Solving for <math> \cos \alpha </math> we get <math> \cos \alpha =\pm \sqrt{\frac{1 + \cos 2\alpha}2}</math> where we look at the quadrant of <math> \alpha </math> to decide if it's positive or negative.  Likewise, we can use the fact that <math> \cos 2\alpha = 1 - 2\sin^2 \alpha </math> to find a half angle identity for sine.  Then, to find a half angle identity for tangent, we just use the fact that <math> \tan \frac x2 =\frac{\sin \frac x2}{\cos \frac x2} </math> and plug in the half angle identities for sine and cosine.
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To summarize:
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{| style="height:150px; margin: 1em auto 1em auto"
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|-
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| <math> \sin \frac{\theta}2 = \pm \sqrt{\frac{1-\cos \theta}2} </math>
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|-
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| <math> \cos \frac{\theta}2 = \pm \sqrt{\frac{1+\cos \theta}2} </math>
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|-
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| <math> \tan \frac{\theta}2 = \pm \sqrt{\frac{1-\cos \theta}{1+\cos \theta}} </math>
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|}
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== Even-Odd Identities ==
 
== Even-Odd Identities ==
  
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* <math>\cos \theta + \cos \gamma = 2 \cos \frac{\theta+\gamma}2 \cos \frac{\theta-\gamma}2</math>
 
* <math>\cos \theta + \cos \gamma = 2 \cos \frac{\theta+\gamma}2 \cos \frac{\theta-\gamma}2</math>
 
* <math>\cos \theta - \cos \gamma = -2 \sin \frac{\theta+\gamma}2 \sin \frac{\theta-\gamma}2</math>
 
* <math>\cos \theta - \cos \gamma = -2 \sin \frac{\theta+\gamma}2 \sin \frac{\theta-\gamma}2</math>
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== Law of Sines ==
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The extended [[Law of Sines]] states
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<center><math> \frac a{\sin A} = \frac b{\sin B} = \frac c{\sin C} = 2R.</math></center>
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== Law of Cosines ==
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The [[Law of Cosines]] states
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<center><math> a^2 = b^2 + c^2 - 2bc\cos A. </math></center>
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== Law of Tangents ==
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The [[Law of Tangents]] states
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<center><math> \frac{b - c}{b + c} = \frac{\tan\frac 12(B-C)}{\tan \frac 12(B+C)}.</math></center>
  
 
== Other Identities ==
 
== Other Identities ==

Revision as of 11:51, 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/Subtraction Identities

Once we have formulas for angle addition, angle subtraction is rather easy to derive. For example, we just look at $\sin(\alpha+(-\beta))$ and we can derive the sine angle subtraction formula using the sine angle addition formula.

$\sin(\alpha + \beta) = \sin \alpha\cos \beta +\sin \beta \cos \alpha$ $\sin(\alpha - \beta) = \sin \alpha \cos \beta - \sin \beta \cos \alpha$
$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ $\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$
$\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1-\tan \alpha \tan \beta}$ $\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1+\tan \alpha \tan \beta}$

Double Angle Identities

Double angle identities are easily derived from the angle addition formulas by just letting $\alpha = \beta$. Doing so yields:

$\sin 2\alpha$ = $2\sin \alpha \cos \alpha$
$\cos 2\alpha$ = $\cos^2 \alpha - \sin^2 \alpha$
= $2\cos^2 \alpha - 1$
= $1-2\sin^2 \alpha$
$\tan 2\alpha$ = $\frac{2\tan \alpha}{1-\tan^2\alpha}$

Half Angle Identities

Using the double angle identities, we can now derive half angle identities. The double angle formula for cosine tells us $\cos 2\alpha = 2\cos^2 \alpha - 1$. Solving for $\cos \alpha$ we get $\cos \alpha =\pm \sqrt{\frac{1 + \cos 2\alpha}2}$ where we look at the quadrant of $\alpha$ to decide if it's positive or negative. Likewise, we can use the fact that $\cos 2\alpha = 1 - 2\sin^2 \alpha$ to find a half angle identity for sine. Then, to find a half angle identity for tangent, we just use the fact that $\tan \frac x2 =\frac{\sin \frac x2}{\cos \frac x2}$ and plug in the half angle identities for sine and cosine.

To summarize:

$\sin \frac{\theta}2 = \pm \sqrt{\frac{1-\cos \theta}2}$
$\cos \frac{\theta}2 = \pm \sqrt{\frac{1+\cos \theta}2}$
$\tan \frac{\theta}2 = \pm \sqrt{\frac{1-\cos \theta}{1+\cos \theta}}$

Even-Odd Identities

Prosthaphaeresis Identities

(Otherwise known as sum-to-product identities)

  • $\sin \theta \pm \sin \gamma = 2 \sin \frac{\theta\pm \gamma}2 \cos \frac{\theta\mp \gamma}2$
  • $\cos \theta + \cos \gamma = 2 \cos \frac{\theta+\gamma}2 \cos \frac{\theta-\gamma}2$
  • $\cos \theta - \cos \gamma = -2 \sin \frac{\theta+\gamma}2 \sin \frac{\theta-\gamma}2$

Law of Sines

The extended Law of Sines states

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

Law of Cosines

The Law of Cosines states

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

Law of Tangents

The Law of Tangents states

$\frac{b - c}{b + c} = \frac{\tan\frac 12(B-C)}{\tan \frac 12(B+C)}.$

Other Identities

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

See also

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