Difference between revisions of "Trigonometric identities"

Revision as of 17:30, 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

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

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

Other Identities

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.

$\sin (\frac{\pi}{2} - x) = \cos (x)$ and $\cos (\frac{\pi}{2} - x) = \sin (x)$
$\tan (\frac{\pi}{2} - x) = \cot (x)$ and $\cot (\frac{\pi}{2} - x) = \tan (x)$
$\csc (\frac{\pi}{2} - x) = \sec (x)$ and $\sec (\frac{\pi}{2} - x) = \csc (x)$

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 are still important, but do not contain enough substance to warrant a section of their own.

$\sin(180-\theta) = \sin(\theta)$
$\cos(180-\theta) = -\cos(\theta)$
$\tan(180-\theta) = -\tan(\theta)$

External Links

Trigonometric Identities

@@ Line 1: / Line 1: @@
 '''Trigonometric Identities''' are used to manipulate [[trigonometry]] [[equation]]s in certain ways.  Here is a list of them:
-== Even-Odd Identities ==
+== Algebraic 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:
+These are the identities that are used when doing algebra with trigonometry.
-* <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>.
-== Pythagorean Identities ==
+=== Pythagorean identities ===
 The Pythagorean identities state that
 * <math>\sin^2x + \cos^2x = 1</math>
@@ Line 18: / Line 10: @@
 * <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 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>.
+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 Addition Identities ==
+=== Angle addition identities ===
 The trigonometric angle addition identities state the following identities:
 * <math>\sin(x + y) = \sin (x) \cos (y) + \cos (x) \sin (y)</math>
@@ Line 47: / Line 41: @@
 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:
 * <math>\sin (2x) = 2\sin (x) \cos (x)</math>
@@ Line 53: / Line 47: @@
 * <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.
 * <math>\cos (2x) = 1 - 2 \sin^2 (x)</math>
@@ Line 62: / Line 56: @@
 * <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:
 * <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>\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 (\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.
 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-Product Identities ==
+=== Sum-to-product identities ===
 * <math>{\sin \theta + \sin \gamma = 2 \sin \frac{\theta + \gamma}2 \cos \frac{\theta - \gamma}2}</math>
@@ Line 78: / Line 72: @@
 * <math>{\cos \theta - \cos \gamma = -2 \sin \frac{\theta+\gamma}2 \sin \frac{\theta-\gamma}2}</math>
-== Law of Sines ==
+=== Product-to-sum identities ===
-{{main|Law of Sines}}
-The extended [[Law of Sines]] states
-*<math>\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R</math>
+== Other Identities ==
-== Law of Cosines ==
+=== Even-odd identities ===
-{{main|Law of Cosines}}
+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:
-The [[Law of Cosines]] states
+* <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>
-*<math>a^2 = b^2 + c^2 - 2bc\cos 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>.
-== Law of Tangents ==
+=== Conversion identities ===
-{{main|Law of Tangents}}
+The following identities are useful when converting trigonometric functions.
-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>\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> \frac{\tan{\left(\frac{A-B}{2}\right)}}{\tan{\left(\frac{A+B}{2}\right)}}=\frac{a-b}{a+b} . </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.
-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
+=== Miscellaneous ===
-<math>\tan(A)\cdot\tan(B)\cdot\tan(C)=\tan(A)+\tan(B)+\tan(C)</math>
+These are the identities that are still important, but do not contain enough substance to warrant a section of their own.
-== 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>\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.
-*<math>\cos(2\theta) = (\cos(\theta) + \sin(\theta))(\cos(\theta) - \sin(\theta))</math>
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