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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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In [[trigonometry]], '''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.
  
−
== Basic Definitions ==
+
== Pythagorean identities ==
−
The six basic trigonometric functions can be defined using a right triangle:
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The Pythagorean identities state that
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<center>[[Image:righttriangle.png]]</center>
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* <math>\sin^2x + \cos^2x = 1</math>
 +
* <math>1 + \cot^2x = \csc^2x</math>
 +
* <math>\tan^2x + 1 = \sec^2x</math>
  
−
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 <math>\csc</math>).  They are defined as follows:
+
Using the unit circle definition of trigonometry, because the point <math>(\cos (x), \sin (x))</math> is defined to be on the unit circle, it is a distance one away from the origin. Then by the distance formula, <math>\sin^2x + \cos^2x = 1</math>. 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.
−
{| class="wikitable"
 
−
|+ Basic Definitions
 
−
|- <!-- Start of a new row -->
 
−
| <math>\sin A = \frac ac</math>  || <math>\csc A = \frac ca</math> || <math> \cos A = \frac bc</math> || <math>\sec A = \frac cb</math> || <math> \tan A = \frac ab</math> || <math> \cot A = \frac ba</math>
 
−
|}
 
  
−
== Even-Odd Identities ==
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== Angle addition identities ==
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*<math>\sin (-\theta) = -\sin (\theta) </math>
+
The trigonometric angle addition identities state the following identities:
 +
* <math>\sin(x + y) = \sin (x) \cos (y) + \cos (x) \sin (y)</math>
 +
* <math>\cos(x + y) = \cos (x) \cos (y) - \sin (x) \sin (y) </math>
 +
* <math>\tan(x + y) = \frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan (y)} </math>
 +
There are many proofs of these identities. For the sake of brevity, we list only one here.
  
−
*<math>\cos (-\theta) = \cos (\theta) </math>
+
[[Euler's identity]] states that <math>e^{ix} = \cos (x) + i \sin(x)</math>. We have that
 +
<cmath>\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*}</cmath>
 +
By looking at the real and imaginary parts, we derive the sine and cosine angle addition formulas.
  
−
*<math>\tan (-\theta) = -\tan (\theta) </math>
+
To derive the tangent addition formula, we reduce the problem to use sine and cosine, divide both numerator and denominator by <math>\cos (x) \cos (y)</math>, and simplify.
 +
<cmath>\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(y)}}{1 - \frac{\sin (x) \sin(y)}{\cos (x) \cos(y)}} \\
 +
&= \frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan(y)}
 +
\end{align*}</cmath>
 +
as desired.
  
−
*<math>\sec (-\theta) = \sec (\theta) </math>
+
== 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:
 +
* <math>\sin (2x) = 2\sin (x) \cos (x)</math>
 +
* <math>\cos (2x) = \cos^2 (x) - \sin^2 (x)</math>
 +
* <math>\tan (2x) = \frac{2\tan (x)}{1-\tan^2 (x)}</math>
  
−
*<math>\csc (-\theta) = -\csc (\theta) </math>
+
=== 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.
 +
* <math>\cos (2x) = 1 - 2 \sin^2 (x)</math>
 +
* <math>\cos (2x) = 2 \cos^2 (x) - 1</math>
  
−
*<math>\cot (-\theta) = -\cot (\theta) </math>
+
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.
 +
* <math>\sin^2 (x) = \frac{1 - \cos (2x)}{2}</math>
 +
* <math>\cos^2 (x) = \frac{1 + \cos (2x)}{2}</math>
  
−
===Further Conclusions===
+
== Half-angle identities ==
 +
The trigonometric half-angle identities state the following equalities:
 +
* <math>\sin \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos (x)}{2}}</math>
 +
* <math>\cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos (x)}{2}}</math>
 +
* <math>\tan \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos (x)}{1+\cos (x)}} = \frac{\sin (x)}{1 + \cos (x)} = \frac{1-\cos (x)}{\sin (x)}</math>
 +
The plus or minus does not mean that there are two answers, but that the sign of the expression depends on the quadrant in which the angle resides.
  
−
Based on the above identities, we can also claim that
+
Consider 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. As for the tangent identity, divide the sine and cosine half-angle identities.
  
−
*<math>\sin(\cos(-\theta)) = \sin(\cos(\theta))</math>
+
== Product-to-sum identities ==
 +
The product-to-sum identities are as follows:
 +
* <math>\sin (x) \sin (y) = \frac{1}{2} (\cos (x-y) - \cos (x+y))</math>
 +
* <math>\sin (x) \cos (y) = \frac{1}{2} (\sin (x-y) + \sin (x+y))</math>
 +
* <math>\cos (x) \cos (y) = \frac{1}{2} (\cos (x-y) + \cos (x+y))</math>
 +
They can be derived by expanding out <math>\cos (x+y)</math> and <math>\cos (x-y)</math> or <math>\sin (x+y)</math> and <math>\sin(x-y)</math>, then combining them to isolate each term.
  
−
*<math>\cos(\sin(-\theta)) = \cos(-\sin(\theta)) = \cos(\sin(\theta))</math>
+
== Sum-to-product identities ==
 +
Substituting <math>\alpha = x+y</math> and <math>\beta = x-y</math> into the product-to-sum identities yields the sum-to-product identities.
  
−
This is only true when <math>\sin(\theta)</math> is in the domain of <math>\cos(\theta)</math>.
+
* <math>\sin (x) + \sin (y) =  2 \sin \left(\frac{x + y}{2}\right) \cos \left(\frac{x - y}{2}\right)</math>
 +
* <math>\sin (x) - \sin (y) =  2 \sin \left(\frac{x - y}{2}\right) \cos \left(\frac{x + y}{2}\right)</math>
 +
* <math>\cos (x) + \cos (y) =  2 \cos \left(\frac{x + y}{2}\right) \cos \left(\frac{x - y}{2}\right)</math>
 +
* <math>\cos (x) - \cos (y) = -2 \sin \left(\frac{x + y}{2}\right) \sin \left(\frac{x - y}{2}\right)</math>
  
−
== Reciprocal Relations ==
+
== Other identities ==
−
From the first section, it is easy to see that the following hold:
+
Here are some identities that are less significant than those above, but still useful.
  
−
*<math> \sin A = \frac 1{\csc A}</math>  
+
=== Triple-angle identities ===
 +
* <math>\sin 3x = 3\sin x-4\sin^3 x</math>
 +
* <math>\cos 3x = 4\cos^3 x-3\cos x</math>
 +
* <math>\tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}</math>
 +
All of these expansions can be proved using <cmath>\sin 3x = \sin (2x+x)</cmath> trick and perform the angle addition identities. Same for <math>\cos 3x</math> and for <math>\tan 3x</math>.
  
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*<math> \cos A = \frac 1{\sec A}</math>
+
=== Even-odd identities ===
 +
The functions <math>\sin(x)</math>, <math>\tan(x)</math>, <math>\cot(x)</math>, and <math>\csc(x)</math> are odd, while <math>\cos(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>\cot (-x) = -\cot (x) </math>
 +
* <math>\csc (-x) = -\csc (x) </math>
 +
* <math>\sec (-x) =  \sec (x) </math>
  
−
*<math> \tan A = \frac 1{\cot A}</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>.
  
−
Another useful identity that isn't a reciprocal relation is that <math> \tan A =\frac{\sin A}{\cos A} </math>.
+
=== Conversion identities ===
 +
The following identities are useful when converting trigonometric functions.
 +
*<math>\sin (90^{\circ} - x) = \cos (x) \textrm{ and } \cos (90^{\circ} - x) = \sin (x)</math>
 +
*<math>\tan (90^{\circ} - x) = \cot (x) \textrm{ and } \cot (90^{\circ} - x) = \tan (x)</math>
 +
*<math>\csc (90^{\circ} - x) = \sec (x) \textrm{ and } \sec (90^{\circ} - x) = \csc (x)</math>
 +
All of these can be proven via the angle addition identities.
  
−
Note that <math>\sin^{-1} A \neq \csc A</math>; the former refers to the [[inverse trigonometric function]]s.
+
=== Euler's identity ===
 +
[[Euler's identity]] 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 identity is fundamental to the study of complex numbers and is widely considered among the most beautiful formulas in math.
  
−
== Pythagorean Identities ==
+
Similar to the derivation of the product-to-sum identities, we can isolate sine and cosine by comparing <math>e^{ix}</math> and <math>e^{-ix}</math>, which yields the following identities:
−
Using the [[Pythagorean Theorem]] on our triangle above, we know 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 A + \cos^2 A =1 </math>.  Dividing by <math> a^2 </math> or <math> b^2 </math> instead produces two other similar identities. The Pythagorean Identities are listed below:
+
*<math>\cos (x) = \frac{e^{ix} + e^{-ix}}{2}</math>
 +
*<math>\sin (x) = \frac{e^{ix} - e^{-ix}}{2i}</math>
 +
They can also be derived by computing <math>\textrm{Re} (e^{ix})</math> and <math>\textrm{Im} (e^{ix})</math>. These expressions are occasionally used to define the trigonometric functions.
  
−
*<math>\sin^2x + \cos^2x = 1</math>
+
=== Miscellaneous ===
−
*<math>1 + \cot^2x = \csc^2x</math>
+
These are the identities that are not substantial enough to warrant a section of their own.
−
*<math>\tan^2x + 1 = \sec^2x</math>
+
*<math>\sin (180^{\circ} - x) =  \sin (x) \textrm{ and } \csc (180^{\circ} - x) = \csc (x)</math>
 +
*<math>\cos (180^{\circ} - x) = -\cos (x) \textrm{ and } \sec (180^{\circ} - x) = -\sec (x)</math>
 +
*<math>\tan (180^{\circ} - x) = -\tan (x) \textrm{ and } \cot (180^{\circ} - x) = -\cot (x)</math>
 +
* <math>\sin (x) \cos(x) = \frac{\sin (2x)}{2}</math>
  
−
(Note that the last two are easily derived by dividing the first by <math>\sin^2x</math> and <math>\cos^2x</math>, respectively.)
+
== Resources ==
 +
* [http://www.sosmath.com/trig/Trig5/trig5/trig5.html Table of trigonometric identities]
 +
* [https://mathwithtimmy.files.wordpress.com/2017/06/trig-identities.pdf List of Trigonometric Identities]
  
−
== Angle Addition/Subtraction Identities ==
+
== See also ==
−
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 \pm \beta) = \sin \alpha\cos \beta \pm\sin \beta \cos \alpha</math>
 
−
*<math> \cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \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>.
 
−
 
 
−
<math>\cos (\alpha + \beta)</math>
 
−
 
 
−
<math> = \sin((90 -\alpha) - \beta) </math><math>= \sin (90- \alpha) \cos (\beta) - \sin ( \beta) \cos (90- \alpha) </math>
 
−
 
 
−
<math>=\cos \alpha \cos \beta - \sin \beta \sin \alpha </math>
 
−
 
 
−
== Double Angle Identities ==
 
−
Double angle identities are easily derived from the angle addition formulas by just letting <math> \alpha = \beta </math>.  Doing so yields:
 
−
 
 
−
<cmath>\begin{eqnarray*}
 
−
\sin 2\alpha &=& 2\sin \alpha \cos \alpha\\
 
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\cos 2\alpha  &=& \cos^2 \alpha - \sin^2 \alpha\\
 
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&=& 2\cos^2 \alpha - 1\\
 
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&=& 1-2\sin^2 \alpha\\
 
−
\tan 2\alpha  &=& \frac{2\tan \alpha}{1-\tan^2\alpha} \end{eqnarray*}</cmath>
 
−
 
 
−
=Further Conclusions=
 
−
 
 
−
We can see from the above that
 
−
 
 
−
*<math>\csc(2a) = \frac{\csc(a)\sec(a)}{2}</math>
 
−
*<math>\sec(2a) = \frac{1}{2\cos^2(a) - 1} = \frac{1}{\cos^2(a) - \sin^2(a)} = \frac{1}{1 - 2\sin^2(a)}</math>
 
−
*<math>\cot(2a) = \frac{1 - \tan^2(a)}{2\tan(a)}</math>
 
−
 
 
−
== Half Angle Identities ==
 
−
Using the double angle identities, we can derive half angle identities.  The double angle formula for cosine tells us <math>\cos 2\alpha = 2\cos^2 \alpha - 1 </math>.  Solving for <math>\cos \alpha </math> we get <math>\cos \alpha =\pm \sqrt{\frac{1 + \cos 2\alpha}2}</math> where we look at the quadrant of <math>\alpha </math> to decide if it's positive or negative.  Likewise, we can use the fact that <math>\cos 2\alpha = 1 - 2\sin^2 \alpha </math> to find a half angle identity for sine.  Then, to find a half angle identity for tangent, we just use the fact that <math>\tan \frac x2 =\frac{\sin \frac x2}{\cos \frac x2} </math> and plug in the half angle identities for sine and cosine.
 
−
 
 
−
To summarize:
 
−
 
 
−
*<math> \sin \frac{\theta}2 = \pm \sqrt{\frac{1-\cos \theta}2} </math>
 
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*<math> \cos \frac{\theta}2 = \pm \sqrt{\frac{1+\cos \theta}2} </math>
 
−
*<math> \tan \frac{\theta}2 = \pm \sqrt{\frac{1-\cos \theta}{1+\cos \theta}} </math>
 
−
 
 
−
== Prosthaphaeresis Identities ==
 
−
(Otherwise known as sum-to-product identities)
 
−
 
 
−
* <math>\sin \theta \pm \sin \gamma = 2 \sin \frac{\theta\pm \gamma}2 \cos \frac{\theta\mp \gamma}2</math>
 
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* <math>\cos \theta + \cos \gamma = 2 \cos \frac{\theta+\gamma}2 \cos \frac{\theta-\gamma}2</math>
 
−
* <math>\cos \theta - \cos \gamma = -2 \sin \frac{\theta+\gamma}2 \sin \frac{\theta-\gamma}2</math>
 
−
 
 
−
== Law of Sines ==
 
−
{{main|Law of Sines}}
 
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The extended [[Law of Sines]] states
 
−
 
 
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*<math>\frac a{\sin A} = \frac b{\sin B} = \frac c{\sin C} = 2R.</math>
 
−
 
 
−
== Law of Cosines ==
 
−
{{main|Law of Cosines}}
 
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The [[Law of Cosines]] states
 
−
 
 
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*<math>a^2 = b^2 + c^2 - 2bc\cos A. </math>
 
−
 
 
−
== Law of Tangents ==
 
−
{{main|Law of Tangents}}
 
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The [[Law of Tangents]] states that if <math>A</math> and <math>B</math> are angles in a triangle opposite sides <math>a</math> and <math>b</math> respectively, then
 
−
 
 
−
<math> \frac{\tan{\left(\frac{A-B}{2}\right)}}{\tan{\left(\frac{A+B}{2}\right)}}=\frac{a-b}{a+b} . </math>
 
−
 
 
−
A further extension of the [[Law of Tangents]] states that if <math>A</math>, <math>B</math>, and <math>C</math> are angles in a triangle, then
 
−
<math>\tan(A)\cdot\tan(B)\cdot\tan(C)=\tan(A)+\tan(B)+\tan(C)</math>
 
−
 
 
−
== Other Identities ==
 
−
*<math>\sin(90-\theta) = \cos(\theta)</math>
 
−
*<math>\cos(90-\theta)=\sin(\theta)</math>
 
−
*<math>\tan(90-\theta)=\cot(\theta)</math>
 
−
*<math>\sin(180-\theta) = \sin(\theta)</math>
 
−
*<math>\cos(180-\theta) = -\cos(\theta)</math>
 
−
*<math>\tan(180-\theta) = -\tan(\theta)</math>
 
−
*<math>e^{i\theta} = \cos \theta + i\sin \theta</math> (This is also written as <math>\text{cis }\theta</math>)
 
−
*<math>|1-e^{i\theta}|=2\sin\frac{\theta}{2}</math>
 
−
*<math>\left(\tan\theta + \sec\theta\right)^2 = \frac{1 + \sin\theta}{1 - \sin\theta}</math>
 
−
*<math>\sin(\theta) = \cos(\theta)\tan(\theta)</math>
 
−
*<math>\cos(\theta) = \frac{\sin(\theta)}{\tan(\theta)}</math>
 
−
*<math>\sec(\theta) = \frac{\tan(\theta)}{\sin(\theta)}</math>
 
−
*<math>\sin^2(\theta) + \cos^2(\theta) + \tan^2(\theta) = \sec^2(\theta)</math>
 
−
*<math>\sin^2(\theta) + \cos^2(\theta) + \cot^2(\theta) = \csc^2(\theta)</math>
 
−
 
 
−
The two identities right above here were based on identites others posted on this site with a substitution.
 
−
 
 
−
*<math>\cos(2\theta) = (\cos(\theta) + \sin(\theta))(\cos(\theta) - \sin(\theta))</math>
 
−
 
 
−
==See also==
 
 
* [[Trigonometry]]
 
* [[Trigonometry]]
 
* [[Trigonometric substitution]]
 
* [[Trigonometric substitution]]
−
*
+
* [[Proofs of trig identities]]
−
 
 
−
==External Links==
 
−
[http://www.sosmath.com/trig/Trig5/trig5/trig5.html Trigonometric Identities]
 
  
 
[[Category:Trigonometry]]
 
[[Category:Trigonometry]]

Latest revision as of 20:55, 20 January 2024

In trigonometry, 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 unit circle definition of trigonometry, because the point $(\cos (x), \sin (x))$ is defined to be on the unit circle, it is a distance one away from the origin. Then by the distance formula, $\sin^2x + \cos^2x = 1$. 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 the problem to use 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(y)}}{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:

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

The plus or minus does not mean that there are two answers, but that the sign of the expression depends on the quadrant in which the angle resides.

Consider 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. As for the tangent identity, divide the sine and cosine half-angle identities.

Product-to-sum identities

The product-to-sum identities are as follows:

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

They can be derived by expanding out $\cos (x+y)$ and $\cos (x-y)$ or $\sin (x+y)$ and $\sin(x-y)$, then combining them to isolate each term.

Sum-to-product identities

Substituting $\alpha = x+y$ and $\beta = x-y$ into the product-to-sum identities yields the sum-to-product identities.

  • $\sin (x) + \sin (y) = 2 \sin \left(\frac{x + y}{2}\right) \cos \left(\frac{x - y}{2}\right)$
  • $\sin (x) - \sin (y) = 2 \sin \left(\frac{x - y}{2}\right) \cos \left(\frac{x + y}{2}\right)$
  • $\cos (x) + \cos (y) = 2 \cos \left(\frac{x + y}{2}\right) \cos \left(\frac{x - y}{2}\right)$
  • $\cos (x) - \cos (y) = -2 \sin \left(\frac{x + y}{2}\right) \sin \left(\frac{x - y}{2}\right)$

Other identities

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

Triple-angle identities

  • $\sin 3x = 3\sin x-4\sin^3 x$
  • $\cos 3x = 4\cos^3 x-3\cos x$
  • $\tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}$

All of these expansions can be proved using \[\sin 3x = \sin (2x+x)\] trick and perform the angle addition identities. Same for $\cos 3x$ and for $\tan 3x$.

Even-odd identities

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

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

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.

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

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

Euler's identity

Euler's identity 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 identity is fundamental to the study of complex numbers and is widely considered among the most beautiful formulas in math.

Similar to the derivation of the product-to-sum identities, we can isolate sine and cosine by comparing $e^{ix}$ and $e^{-ix}$, which yields the following identities:

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

They can also be derived by computing $\textrm{Re} (e^{ix})$ and $\textrm{Im} (e^{ix})$. These expressions are occasionally used to define the trigonometric functions.

Miscellaneous

These are the identities that are not substantial enough to warrant a section of their own.

  • $\sin (180^{\circ} - x) = \sin (x) \textrm{ and } \csc (180^{\circ} - x) = \csc (x)$
  • $\cos (180^{\circ} - x) = -\cos (x) \textrm{ and } \sec (180^{\circ} - x) = -\sec (x)$
  • $\tan (180^{\circ} - x) = -\tan (x) \textrm{ and } \cot (180^{\circ} - x) = -\cot (x)$
  • $\sin (x) \cos(x) = \frac{\sin (2x)}{2}$

Resources

See also

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