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Trigonometric identities

Revision as of 07:48, 24 June 2006 by Joml88 (talk | contribs) (Pythagorean identities)

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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