Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Trigonometric identities"
m
 
(28 intermediate revisions by 8 users not shown)
Line 1: Line 1:
−
'''Trigonometric Identities''' are used to manipulate [[trigonometry]] [[equation]]s in certain ways. Here is a list of them:
+
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.
  
−
== Even-Odd Identities ==
+
== Pythagorean 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 flipping 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>.
 
−
 
 
−
== 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>
Line 18: Line 7:
 
* <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 unit circle definition of trigonometry, it's extremely easy to see that <math>\sin^2 x+ \cos^2 x =1 </math>.
+
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.
−
 
 
−
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.
 
  
−
== Angle Addition Identities ==
+
== 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>
Line 38: Line 25:
 
By looking at the real and imaginary parts, we derive the sine and cosine angle addition formulas.
 
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 <math>\cos (x) \cos (y)</math>, and simplify.
+
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*}
 
<cmath>\begin{align*}
 
\tan (x+y) &= \frac{\sin (x+y)}{\cos (x+y)} \\
 
\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{\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{\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)}
 
&= \frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan(y)}
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 
as desired.
 
as desired.
  
−
== Double-Angle Identities ==
+
== 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>
 
* <math>\cos (2x) = \cos^2 (x) - \sin^2 (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>\tan (2x) = \frac{2\tan (x)}{1-\tan^2 (x)}</math>
  
−
=== Cosine Double-Angle ===
+
=== 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>
Line 62: Line 49:
 
* <math>\cos^2 (x) = \frac{1 + \cos (2x)}{2}</math>
 
* <math>\cos^2 (x) = \frac{1 + \cos (2x)}{2}</math>
  
−
== Half-Angle Identities ==
+
== 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 \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 - \cos (x)}{2}}</math>
−
* <math>\cos (\frac{x}{2}) = \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 (\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>
+
* <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 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 does not mean that there are two answers, but that the sign of the expression depends on the quadrant in which the angle resides.
  
−
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.
+
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.
  
−
== Sum-to-Product Identities ==
+
== 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>{\sin \theta + \sin \gamma = 2 \sin \frac{\theta + \gamma}2 \cos \frac{\theta - \gamma}2}</math>
+
== Sum-to-product identities ==
−
* <math>{\sin \theta - \sin \gamma = 2 \sin \frac{\theta - \gamma}2 \cos \frac{\theta + \gamma}2}</math>
+
Substituting <math>\alpha = x+y</math> and <math>\beta = x-y</math> into the product-to-sum identities yields the sum-to-product identities.
−
* <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 ==
+
* <math>\sin (x) + \sin (y) = 2 \sin \left(\frac{x + y}{2}\right) \cos \left(\frac{x - y}{2}\right)</math>
−
{{main|Law of Sines}}
+
* <math>\sin (x) - \sin (y) = 2 \sin \left(\frac{x - y}{2}\right) \cos \left(\frac{x + y}{2}\right)</math>
−
The extended [[Law of Sines]] states
+
* <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>
  
−
*<math>\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R</math>
+
== Other identities ==
 +
Here are some identities that are less significant than those above, but still useful.
  
−
== Law of Cosines ==
+
=== Triple-angle identities ===
−
{{main|Law of Cosines}}
+
* <math>\sin 3x = 3\sin x-4\sin^3 x</math>
−
The [[Law of Cosines]] states
+
* <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>.
  
−
*<math>a^2 = b^2 + c^2 - 2bc\cos 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>
  
−
== Law of Tangents ==
+
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>.
−
{{main|Law of Tangents}}
 
−
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>
+
=== 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.
  
−
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
+
=== Euler's identity ===
−
<math>\tan(A)\cdot\tan(B)\cdot\tan(C)=\tan(A)+\tan(B)+\tan(C)</math>
+
[[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.
  
−
== Other 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:
−
*<math>\sin(90-\theta) = \cos(\theta)</math>
+
*<math>\cos (x) = \frac{e^{ix} + e^{-ix}}{2}</math>
−
*<math>\cos(90-\theta)=\sin(\theta)</math>
+
*<math>\sin (x) = \frac{e^{ix} - e^{-ix}}{2i}</math>
−
*<math>\tan(90-\theta)=\cot(\theta)</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(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>\arctan(x) + \arctan(y) = \arctan \left( \dfrac{x+y}{1-xy} \right)</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 above are derived from the Pythagorean Identities.
+
=== Miscellaneous ===
 +
These are the identities that are not substantial enough to warrant a section of their own.
 +
*<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>
  
−
*<math>\cos(2\theta) = (\cos(\theta) + \sin(\theta))(\cos(\theta) - \sin(\theta))</math>
+
== 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]
  
−
==See also==
+
== 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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Trigonometric_identities&oldid=211721"