Difference between revisions of "Trigonometric identities"

Revision as of 18:52, 11 March 2007

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:

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 $\displaystyle c^2$ we get $\displaystyle \left(\frac ac\right)^2 + \left(\frac bc\right)^2 = 1$ which is just $\displaystyle \sin^2 A + \cos^2 A =1$ . Dividing by $\displaystyle a^2$ or $\displaystyle b^2$ instead produces two other similar identities. The Pythagorean Identities are listed below:

$\displaystyle \sin^2x + \cos^2x = 1$

$\displaystyle 1 + \cot^2x = \csc^2x$

$\displaystyle \tan^2x + 1 = \sec^2x$

(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.

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

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

Double Angle Identities

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

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

Half Angle Identities

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

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

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

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

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

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

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

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

The extended Law of Sines states

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

Law of Cosines

The Law of Cosines states

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

Law of Tangents

The Law of Tangents states

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

Other Identities

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

@@ Line 58: / Line 58: @@
 |}
 We can prove <math> \displaystyle  \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta </math> easily by using <math> \displaystyle  \sin(\alpha + \beta) = \sin \alpha\cos \beta +\sin \beta \cos \alpha</math> and <math>\sin(x)=\cos(90-x)</math>
-<math>\displaystyle \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 </math>
+<math>\displaystyle \cos (\alpha + \beta) = \sin((90 -\alpha) - \beta) </math><math>\displaystyle = \sin (90- \alpha) \cos (\beta) - \sin ( \beta) \cos (90- \alpha) </math><math>=\displaystyle \cos \alpha \cos \beta - \sin \beta \sin \alpha </math>
 == Double Angle Identities ==

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