Difference between revisions of "Trigonometric identities"

m (Law of Tangents)
(Angle Addition/Subtraction Identities)
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Once we have formulas for angle addition, angle subtraction is rather easy to derive.  For example, we just look at <math> \sin(\alpha+(-\beta))</math> and we can derive the sine angle subtraction formula using the sine angle addition formula.
 
Once we have formulas for angle addition, angle subtraction is rather easy to derive.  For example, we just look at <math> \sin(\alpha+(-\beta))</math> and we can derive the sine angle subtraction formula using the sine angle addition formula.
  
*<math> \sin(\alpha + \beta) = \sin \alpha\cos \beta +\sin \beta \cos \alpha</math> || <math>\sin(\alpha - \beta) = \sin \alpha \cos \beta - \sin \beta \cos \alpha</math>
+
*<math> \sin(\alpha \pm \beta) = \sin \alpha\cos \beta \pm\sin \beta \cos \alpha</math>
*<math> \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta </math> || <math>\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta</math>
+
*<math> \cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta </math>
*<math>\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1-\tan \alpha \tan \beta} </math> || <math>\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1+\tan \alpha \tan \beta} </math>
+
*<math>\tan(\alpha \pm \beta) = \frac{\tan \alpha \pm \tan \beta}{1\mp\tan \alpha \tan \beta} </math>
  
 
We can prove <math> \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta </math> easily by using <math> \sin(\alpha + \beta) = \sin \alpha\cos \beta +\sin \beta \cos \alpha</math> and <math>\sin(x)=\cos(90-x)</math>.
 
We can prove <math> \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta </math> easily by using <math> \sin(\alpha + \beta) = \sin \alpha\cos \beta +\sin \beta \cos \alpha</math> and <math>\sin(x)=\cos(90-x)</math>.
 
  
 
<math>\cos (\alpha + \beta)</math>
 
<math>\cos (\alpha + \beta)</math>

Revision as of 22:22, 11 March 2009

Trigonometric identities are used to manipulate trigonometry 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:

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

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.

  • $\sin(\alpha \pm \beta) = \sin \alpha\cos \beta \pm\sin \beta \cos \alpha$
  • $\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$
  • $\tan(\alpha \pm \beta) = \frac{\tan \alpha \pm \tan \beta}{1\mp\tan \alpha \tan \beta}$

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. ! Missing \endgroup inserted.)

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 that if $A$ and $B$ are angles in a triangle opposite sides $a$ and $b$ respectively, then

$\frac{\tan{\left(\frac{A-B}{2}\right)}}{\tan{\left(\frac{A+B}{2}\right)}}=\frac{a-b}{a+b} .$

Other Identities

  • $e^{i\theta} = \cos \theta + i\sin \theta$ (This is also written as $\text{cis }\theta$)
  • $|1-e^{i\theta}|=2\sin\frac{\theta}{2}$

See also

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