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Difference between revisions of "Trigonometric identities"
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'''Trigonometric Identities''' are used to manipulate [[trigonometry]] [[equation]]s in certain ways. Here is a list of them:
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'''Trigonometric identities''' are equations involving [[trigonometric functions]] that are true for all input values. Trigonometric functions have an abundance of identities, of which only the most widely used are included in this article.
  
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== Algebraic identities ==
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== Pythagorean identities ==
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These are the identities that are used when doing algebra with trigonometry.
 
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=== Pythagorean identities ===
 
 
The Pythagorean identities state that
 
The Pythagorean identities state that
 
* <math>\sin^2x + \cos^2x = 1</math>
 
* <math>\sin^2x + \cos^2x = 1</math>
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To derive the other two Pythagorean identities, divide by either <math>\sin^2 (x)</math> or <math>\cos^2 (x)</math> and substitute the respective trigonometry in place of the ratios to obtain the desired result.
 
To derive the other two Pythagorean identities, divide by either <math>\sin^2 (x)</math> or <math>\cos^2 (x)</math> and substitute the respective trigonometry in place of the ratios to obtain the desired result.
  
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=== Angle addition identities ===
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== Angle addition identities ==
 
The trigonometric angle addition identities state the following identities:
 
The trigonometric angle addition identities state the following identities:
 
* <math>\sin(x + y) = \sin (x) \cos (y) + \cos (x) \sin (y)</math>
 
* <math>\sin(x + y) = \sin (x) \cos (y) + \cos (x) \sin (y)</math>
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as desired.
 
as desired.
  
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=== Double-angle identities ===
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== Double-angle identities ==
 
The trigonometric double-angle identities are easily derived from the angle addition formulas by just letting <math>x = y </math>.  Doing so yields:
 
The trigonometric double-angle identities are easily derived from the angle addition formulas by just letting <math>x = y </math>.  Doing so yields:
 
* <math>\sin (2x) = 2\sin (x) \cos (x)</math>
 
* <math>\sin (2x) = 2\sin (x) \cos (x)</math>
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Take the two expressions listed in the cosine double-angle section for <math>\sin^2 (x)</math> and <math>\cos^2 (x)</math>, and substitute <math>\frac{1}{2} x</math> instead of <math>x</math>. Taking the square root then yields the desired half-angle identities for sine and cosine.
 
Take the two expressions listed in the cosine double-angle section for <math>\sin^2 (x)</math> and <math>\cos^2 (x)</math>, and substitute <math>\frac{1}{2} x</math> instead of <math>x</math>. Taking the square root then yields the desired half-angle identities for sine and cosine.
  
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=== Sum-to-product identities ===
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== Sum-to-product identities ==
  
 
* <math>{\sin \theta + \sin \gamma = 2 \sin \frac{\theta + \gamma}2 \cos \frac{\theta - \gamma}2}</math>
 
* <math>{\sin \theta + \sin \gamma = 2 \sin \frac{\theta + \gamma}2 \cos \frac{\theta - \gamma}2}</math>
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* <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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=== Product-to-sum identities ===
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== Product-to-sum identities ==
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Coming soon
  
 
== Other Identities ==
 
== Other Identities ==
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Here are some identities that are less significant than those above, but still useful.
  
 
=== Even-odd identities ===
 
=== Even-odd identities ===
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=== Conversion identities ===
 
=== Conversion identities ===
 
The following identities are useful when converting trigonometric functions.
 
The following identities are useful when converting trigonometric functions.
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*<math>\sin (\frac{\pi}{2} - x) = \cos (x)</math> and <math>\cos (\frac{\pi}{2} - x) = \sin (x)</math>
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*<math>\sin (90^{\circ} - x) = \cos (x)</math> and <math>\cos (90^{\circ} - x) = \sin (x)</math>
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*<math>\tan (\frac{\pi}{2} - x) = \cot (x)</math> and <math>\cot (\frac{\pi}{2} - x) = \tan (x)</math>
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*<math>\tan (90^{\circ} - x) = \cot (x)</math> and <math>\cot (90^{\circ} - x) = \tan (x)</math>
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*<math>\csc (\frac{\pi}{2} - x) = \sec (x)</math> and <math>\sec (\frac{\pi}{2} - x) = \csc (x)</math>
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*<math>\csc (90^{\circ} - x) = \sec (x)</math> and <math>\sec (90^{\circ} - x) = \csc (x)</math>
 
All of these can be proven via the angle addition identities.
 
All of these can be proven via the angle addition identities.
  
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=== Miscellaneous ===
 
=== Miscellaneous ===
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These are the identities that are still important, but do not contain enough substance to warrant a section of their own.
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These are the identities that do not contain enough substance to warrant a section of their own.
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*<math>\sin(180-\theta) = \sin(\theta)</math>
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*<math>\sin (180^{\circ} - x) = \sin (x)</math> and *<math>\csc (180^{\circ} - x) = \csc (x)</math>
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*<math>\cos(180-\theta) = -\cos(\theta)</math>
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*<math>\cos (180^{\circ} - x) = -\cos (x)</math> and *<math>\sec (180^{\circ} - x) = -\sec (x)</math>
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*<math>\tan(180-\theta) = -\tan(\theta)</math>
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*<math>\tan (180^{\circ} - x) = -\tan (x)</math> and <math>\cot (180^{\circ} - x) = -\cot (x)</math>
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==See also==
 
==See also==

Revision as of 18:37, 30 May 2021

Trigonometric identities are equations involving trigonometric functions that are true for all input values. Trigonometric functions have an abundance of identities, of which only the most widely used are included in this article.

Pythagorean identities

The Pythagorean identities state that

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

Using the ratio definition of trigonometry, we apply Pythagorean Theorem on our triangle above to get that $a^2 + b^2 = c^2$. If we divide by $c^2$, we get $\left(\frac{a}{c}\right)^2 + \left(\frac{b}{c}\right)^2 = 1$, which is just $\sin^2 x + \cos^2 x =1$.

If one uses the unit circle definition of trigonometry, it's extremely easy to see that $\sin^2 x+ \cos^2 x =1$ by the Pythagorean theorem.

To derive the other two Pythagorean identities, divide by either $\sin^2 (x)$ or $\cos^2 (x)$ and substitute the respective trigonometry in place of the ratios to obtain the desired result.

Angle addition identities

The trigonometric angle addition identities state the following identities:

  • $\sin(x + y) = \sin (x) \cos (y) + \cos (x) \sin (y)$
  • $\cos(x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$
  • $\tan(x + y) = \frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan (y)}$

There are many proofs of these identities. For the sake of brevity, we list only one here.

Euler's identity states that $e^{ix} = \cos (x) + i \sin(x)$. We have that \begin{align*} \cos (x+y) + i \sin (x+y) &= e^{i(x+y)} \\ &= e^{ix} \cdot e^{iy} \\ &= (\cos (x) + i \sin (x))(\cos (y) + i \sin (y)) \\ &= (\cos (x) \cos (y) - \sin (x) \sin(y)) + i(\sin (x) \cos(y) + \cos(x) \sin(y)) \end{align*} By looking at the real and imaginary parts, we derive the sine and cosine angle addition formulas.

To derive the tangent addition formula, we reduce tangent to sine and cosine, divide both numerator and denominator by $\cos (x) \cos (y)$, and simplify. \begin{align*} \tan (x+y) &= \frac{\sin (x+y)}{\cos (x+y)} \\ &= \frac{\sin (x) \cos(y) + \cos(x) \sin(y)}{\cos (x) \cos (y) - \sin (x) \sin(y)} \\ &= \frac{\frac{\sin(x)}{\cos(x)} + \frac{\sin(y)}{\cos(x)}}{1 - \frac{\sin (x) \sin(y)}{\cos (x) \cos(y)}} \\ &= \frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan(y)} \end{align*} as desired.

Double-angle identities

The trigonometric double-angle identities are easily derived from the angle addition formulas by just letting $x = y$. Doing so yields:

  • $\sin (2x) = 2\sin (x) \cos (x)$
  • $\cos (2x) = \cos^2 (x) - \sin^2 (x)$
  • $\tan (2x) = \frac{2\tan (x)}{1-\tan^2 (x)}$

Cosine double-angle identity

Here are two equally useful forms of the cosine double-angle identity. Both are derived via the Pythagorean identity on the cosine double-angle identity given above.

  • $\cos (2x) = 1 - 2 \sin^2 (x)$
  • $\cos (2x) = 2 \cos^2 (x) - 1$

In addition, the following identities are useful in integration and in deriving the half-angle identities. They are a simple rearrangement of the two above.

  • $\sin^2 (x) = \frac{1 - \cos (2x)}{2}$
  • $\cos^2 (x) = \frac{1 + \cos (2x)}{2}$

Half-angle identities

The trigonometric half-angle identities state the following equalities:

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

The plus or minus is not saying that there are two answers, but that the sine of the angle depends on the quadrant that the angle is in.

Take the two expressions listed in the cosine double-angle section for $\sin^2 (x)$ and $\cos^2 (x)$, and substitute $\frac{1}{2} x$ instead of $x$. Taking the square root then yields the desired half-angle identities for sine and cosine.

Sum-to-product identities

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

Product-to-sum identities

Coming soon

Other Identities

Here are some identities that are less significant than those above, but still useful.

Even-odd identities

The functions $\sin(x)$, $\tan(x)$, and $\csc(x)$ are odd, while $cos(x)$, $\cot(x)$, and $\sec(x)$ are even. In other words, the six trigonometric functions satisfy the following equalities:

  • $\sin (-x) = -\sin (x)$
  • $\cos (-x) = \cos (x)$
  • $\tan (-x) = -\tan (x)$
  • $\sec (-x) = \sec (x)$
  • $\csc (-x) = -\csc (x)$
  • $\cot (-x) = -\cot (x)$

These are derived by the unit circle definitions of trigonometry. Making any angle negative is the same as reflecting it across the x-axis. This keeps its x-coordinate the same, but makes its y-coordinate negative. Thus, $\sin(-x) = -\sin(x)$ and $\cos(-x) = \cos(x)$.

Conversion identities

The following identities are useful when converting trigonometric functions.

  • $\sin (90^{\circ} - x) = \cos (x)$ and $\cos (90^{\circ} - x) = \sin (x)$
  • $\tan (90^{\circ} - x) = \cot (x)$ and $\cot (90^{\circ} - x) = \tan (x)$
  • $\csc (90^{\circ} - x) = \sec (x)$ and $\sec (90^{\circ} - x) = \csc (x)$

All of these can be proven via the angle addition identities.

Euler's formula

Euler's formula is a formula in complex analysis that connects complex exponentiation with trigonometry. It states that for any real number $x$, \[e^{ix} = \cos (x) + i \sin (x),\] where $e$ is Euler's constant and $i$ is the imaginary unit. Euler's formula is fundamental to the study of complex numbers and is widely considered among the most beautiful formulas in math.

Miscellaneous

These are the identities that do not contain enough substance to warrant a section of their own.

  • $\sin (180^{\circ} - x) = \sin (x)$ and *$\csc (180^{\circ} - x) = \csc (x)$
  • $\cos (180^{\circ} - x) = -\cos (x)$ and *$\sec (180^{\circ} - x) = -\sec (x)$
  • $\tan (180^{\circ} - x) = -\tan (x)$ and $\cot (180^{\circ} - x) = -\cot (x)$


See also

External Links

Trigonometric Identities

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