Revision as of 19:47, 26 October 2007

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

Basic Definitions

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

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:

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

Even-Odd Identities

$\sin (-\theta) = -\sin (\theta)$

$\cos (-\theta) = \cos (\theta)$

$\tan (-\theta) = -\tan (\theta)$

$\csc (-\theta) = -\csc (\theta)$

$\sec (-\theta) = \sec (\theta)$

$\cot (-\theta) = -\cot (\theta)$

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

Note that $\sin^{-1} A \neq \csc A$ ; the former refers to the inverse trigonometric functions.

Pythagorean Identities

Using the Pythagorean Theorem on our triangle above, we know that $a^2 + b^2 = c^2$ . 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:

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

(Note that the second two are easily derived by dividing the first by $\cos^2x$ and $\sin^2x$ )

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.

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

We can prove $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ easily by using $\sin(\alpha + \beta) = \sin \alpha\cos \beta +\sin \beta \cos \alpha$ and $\sin(x)=\cos(90-x)$ .

$\cos (\alpha + \beta)$

$= \sin((90 -\alpha) - \beta)$ $= \sin (90- \alpha) \cos (\beta) - \sin ( \beta) \cos (90- \alpha)$

$=\cos \alpha \cos \beta - \sin \beta \sin \alpha$

Double Angle Identities

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

$\begin{eqnarray*} \sin 2\alpha &=& 2\sin \alpha \cos \alpha\\ \cos 2\alpha &=& \cos^2 \alpha - \sin^2 \alpha\\ &=& 2\cos^2 \alpha - 1\\ &=& 1-2\sin^2 \alpha\\ \tan 2\alpha &=& \frac{2\tan \alpha}{1-\tan^2\alpha}$ (Error compiling LaTeX. Unknown error_msg)

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:

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

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

Main article: 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

Main article: Law of Cosines

The Law of Cosines states

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

Law of Tangents

Main article: 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

$e^{i\pi} = \cos \theta + i\sin \theta$
$|1-e^{i\theta}|=2\sin\frac{\theta}{2}$

@@ Line 1: / Line 1: @@
-'''Trigonometric identities''' are used to manipulate trig equations in certain ways.  Here is a list of them:
+'''Trigonometric identities''' are used to manipulate [[tringonmetry]] [[equation]]s in certain ways.  Here is a list of them:
 == Basic Definitions ==
 The six basic trigonometric functions can be defined using a right triangle:
 <center>[[Image:righttriangle.png]]</center>
 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 <math>\csc</math>).  They are defined as follows:
@@ Line 15: / Line 13: @@
 *<math> \tan A = \frac ab</math>
 *<math> \cot A = \frac ba</math>
+== Even-Odd Identities ==
+*<math>\sin (-\theta) = -\sin (\theta) </math>
+*<math>\cos (-\theta) = \cos (\theta) </math>
+*<math>\tan (-\theta) = -\tan (\theta) </math>
+*<math>\csc (-\theta) = -\csc (\theta) </math>
+*<math>\sec (-\theta) = \sec (\theta) </math>
+*<math>\cot (-\theta) = -\cot (\theta) </math>
 == Reciprocal Relations ==
@@ Line 23: / Line 34: @@
 *<math> \tan A = \frac 1{\cot A}</math>
+Another useful identity that isn't a reciprocal relation is that <math> \tan A =\frac{\sin A}{\cos A} </math>.
-Another useful identity that isn't a reciprocal relation is that <math> \tan A =\frac{\sin A}{\cos A} </math>.
+Note that <math>\sin^{-1} A \neq \csc A</math>; the former refers to the [[inverse trigonometric function]]s.
 == Pythagorean Identities ==
@@ Line 70: / Line 82: @@
 *<math> \tan \frac{\theta}2 = \pm \sqrt{\frac{1-\cos \theta}{1+\cos \theta}} </math>
+== Prosthaphaeresis Identities ==
-== Even-Odd Identities ==
-*<math>\sin (-\theta) = -\sin (\theta) </math>
-*<math>\cos (-\theta) = \cos (\theta) </math>
-*<math>\tan (-\theta) = -\tan (\theta) </math>
-*<math>\csc (-\theta) = -\csc (\theta) </math>
-*<math>\sec (-\theta) = \sec (\theta) </math>
-*<math>\cot (-\theta) = -\cot (\theta) </math>
-==Prosthaphaeresis Identities==
 (Otherwise known as sum-to-product identities)
@@ Line 94: / Line 88: @@
 * <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>
 == Law of Sines ==
@@ Line 116: / Line 108: @@
 == Other Identities =
+*<math>e^{i\pi} = \cos \theta + i\sin \theta</math>
 *<math>|1-e^{i\theta}|=2\sin\frac{\theta}{2}</math>
@@ Line 122: / Line 115: @@
 * [[Trigonometric substitution]]
 * [http://www.sosmath.com/trig/Trig5/trig5/trig5.html Trigonometric Identities]
 [[Category:Trigonometry]]

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Difference between revisions of "Trigonometric identities"