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: | '''Trigonometric Identities''' are used to manipulate [[trigonometry]] [[equation]]s in certain ways. Here is a list of them: | ||
− | == | + | == Algebraic identities == |
− | + | These are the identities that are used when doing algebra with trigonometry. | |
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− | == Pythagorean | + | === 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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* <math>\tan^2x + 1 = \sec^2x</math> | * <math>\tan^2x + 1 = \sec^2x</math> | ||
− | Using the ratio definition of trigonometry, we apply [[Pythagorean Theorem]] on our triangle above to get that <math>a^2 + b^2 = c^2 </math>. If we divide by <math>c^2</math>, we get <math>\left(\frac{a}{c}\right)^2 + \left(\frac{b}{c}\right)^2 = 1 </math>, which is just <math>\sin^2 x + \cos^2 x =1 </math>. | + | Using the ratio definition of trigonometry, we apply [[Pythagorean Theorem]] on our triangle above to get that <math>a^2 + b^2 = c^2 </math>. If we divide by <math>c^2</math>, we get <math>\left(\frac{a}{c}\right)^2 + \left(\frac{b}{c}\right)^2 = 1 </math>, which is just <math>\sin^2 x + \cos^2 x =1 </math>. |
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+ | If one uses the unit circle definition of trigonometry, it's extremely easy to see that <math>\sin^2 x+ \cos^2 x =1 </math> by the [[Pythagorean theorem]]. | ||
− | 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 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. |
− | == Angle | + | === 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. | ||
− | == Double- | + | === 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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* <math>\tan (2x) = \frac{2\tan (x)}{1-\tan^2 (x)}</math> | * <math>\tan (2x) = \frac{2\tan (x)}{1-\tan^2 (x)}</math> | ||
− | === Cosine | + | ==== 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. | 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. | ||
* <math>\cos (2x) = 1 - 2 \sin^2 (x)</math> | * <math>\cos (2x) = 1 - 2 \sin^2 (x)</math> | ||
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* <math>\cos^2 (x) = \frac{1 + \cos (2x)}{2}</math> | * <math>\cos^2 (x) = \frac{1 + \cos (2x)}{2}</math> | ||
− | == Half- | + | === Half-angle identities === |
The trigonometric half-angle identities state the following equalities: | The trigonometric half-angle identities state the following equalities: | ||
* <math>\sin (\frac{x}{2}) = \pm \sqrt{\frac{1 - \cos (x)}{2}}</math> | * <math>\sin (\frac{x}{2}) = \pm \sqrt{\frac{1 - \cos (x)}{2}}</math> | ||
* <math>\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}</math> | * <math>\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}</math> | ||
− | * <math>\tan (\frac{x}{2}) = \pm \sqrt{\frac{1 - \cos (x)}{1+\cos \theta}} = \frac{\sin x}{1 + \cos (x)} = \frac{1-\cos (x)}{\sin ( | + | * <math>\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)}</math> |
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. | 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 <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. | ||
− | == Sum-to- | + | === 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> | ||
− | == | + | === Product-to-sum identities === |
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− | + | == Other Identities == | |
− | == | + | === Even-odd identities === |
− | + | The functions <math>\sin(x)</math>, <math>\tan(x)</math>, and <math>\csc(x)</math> are odd, while <math>cos(x)</math>, <math>\cot(x)</math>, and <math>\sec(x)</math> are even. In other words, the six trigonometric functions satisfy the following equalities: | |
− | + | * <math>\sin (-x) = -\sin (x) </math> | |
+ | * <math>\cos (-x) = \cos (x) </math> | ||
+ | * <math>\tan (-x) = -\tan (x) </math> | ||
+ | * <math>\sec (-x) = \sec (x) </math> | ||
+ | * <math>\csc (-x) = -\csc (x) </math> | ||
+ | * <math>\cot (-x) = -\cot (x) </math> | ||
− | + | 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, <math>\sin(-x) = -\sin(x)</math> and <math>\cos(-x) = \cos(x)</math>. | |
− | == | + | === Conversion identities === |
− | {{ | + | The following identities are useful when converting trigonometric functions. |
− | + | *<math>\sin (\frac{\pi}{2} - x) = \cos (x)</math> and <math>\cos (\frac{\pi}{2} - x) = \sin (x)</math> | |
+ | *<math>\tan (\frac{\pi}{2} - x) = \cot (x)</math> and <math>\cot (\frac{\pi}{2} - x) = \tan (x)</math> | ||
+ | *<math>\csc (\frac{\pi}{2} - x) = \sec (x)</math> and <math>\sec (\frac{\pi}{2} - x) = \csc (x)</math> | ||
+ | All of these can be proven via the angle addition identities. | ||
− | <math> | + | === 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 <math>x</math>, <cmath>e^{ix} = \cos (x) + i \sin (x),</cmath> where <math>e</math> is Euler's constant and <math>i</math> 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 are still important, but do not contain enough substance to warrant a section of their own. | |
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*<math>\sin(180-\theta) = \sin(\theta)</math> | *<math>\sin(180-\theta) = \sin(\theta)</math> | ||
*<math>\cos(180-\theta) = -\cos(\theta)</math> | *<math>\cos(180-\theta) = -\cos(\theta)</math> | ||
*<math>\tan(180-\theta) = -\tan(\theta)</math> | *<math>\tan(180-\theta) = -\tan(\theta)</math> | ||
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==See also== | ==See also== |
Revision as of 18:30, 30 May 2021
Trigonometric Identities are used to manipulate trigonometry equations in certain ways. Here is a list of them:
Algebraic identities
These are the identities that are used when doing algebra with trigonometry.
Pythagorean identities
The Pythagorean identities state that
Using the ratio definition of trigonometry, we apply Pythagorean Theorem on our triangle above to get that . If we divide by
, we get
, which is just
.
If one uses the unit circle definition of trigonometry, it's extremely easy to see that by the Pythagorean theorem.
To derive the other two Pythagorean identities, divide by either or
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:
There are many proofs of these identities. For the sake of brevity, we list only one here.
Euler's identity states that . We have that
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 , and simplify.
as desired.
Double-angle identities
The trigonometric double-angle identities are easily derived from the angle addition formulas by just letting . Doing so yields:
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.
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.
Half-angle identities
The trigonometric half-angle identities state the following equalities:
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 and
, and substitute
instead of
. Taking the square root then yields the desired half-angle identities for sine and cosine.
Sum-to-product identities
Product-to-sum identities
Other Identities
Even-odd identities
The functions ,
, and
are odd, while
,
, and
are even. In other words, the six trigonometric functions satisfy the following equalities:
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, and
.
Conversion identities
The following identities are useful when converting trigonometric functions.
and
and
and
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 ,
where
is Euler's constant and
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 are still important, but do not contain enough substance to warrant a section of their own.