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Difference between revisions of "Angle addition identities"
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<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>
 +
 
<math>\cos(x + y) = \cos (x) \cos (y) - \sin (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>
 
<math>\tan(x + y) = \frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan (y)}</math>
  
 
{{stub}}
 
{{stub}}
 +
 +
==Proofs==
 +
 +
==Proof 1==
 +
 +
<asy>
 +
unitsize(216);
 +
real d = 1/cos(radians(35));
 +
real d1 = d * cos(radians(55));
 +
real d2 = d * sin(radians(55));
 +
pair O = (0,0);
 +
pair A = (cos(radians(20)),0);
 +
pair B = (cos(radians(20)),sin(radians(20)));
 +
pair C = (cos(radians(20)),d2);
 +
pair D = (d1,d2);
 +
draw(O--A--B--O--D--B--O--D--C--B);
 +
dot(O);
 +
dot(B);
 +
dot(A,red);
 +
dot(C,green);
 +
dot(D,blue);
 +
label("O",O,SW);
 +
label("$\alpha$",shift(dir(10)/5)*O);
 +
label("$\beta$",shift(dir(37.5)/5)*O);
 +
label("A",A,SE,red);
 +
label("B",B,E);
 +
label("C",C,NE,green);
 +
label("D",D,dir(122.5),blue);
 +
label("$\cos \alpha$",O--A,S);
 +
label("$\sin \alpha$",A--B,E);
 +
label("1",O--B,dir(302.5));
 +
label("$\frac{\cos \alpha \sin \beta}{\cos \beta}$",B--C,E);
 +
label("$\frac{\sin \alpha \sin \beta}{\cos \beta}$",C--D,N);
 +
label("$\frac{\sin \beta}{\cos \beta}$",B--D,dir(200));
 +
label("$\frac{1}{\cos \beta}$",D--O,dir(325));
 +
</asy>
 +
 +
<math>\fontsize{18}{27}\selectfont \sin (\alpha + \beta ) = \frac{\left( \sin \alpha + \frac{\cos \alpha \sin \beta}{\cos \beta} \right)}{\frac{1}{\cos \beta}} = \cos \beta \times \left( \sin \alpha + \frac{\cos \alpha \sin \beta}{\cos \beta} \right) = \sin \alpha \cos \beta + \cos \alpha \sin \beta</math>
 +
 +
<math>\fontsize{18}{27}\selectfont \cos (\alpha + \beta ) = \frac{\left( \cos \alpha - \frac{\sin \alpha \sin \beta}{\cos \beta} \right)}{\frac{1}{\cos \beta}} = \cos \beta \times \left( \cos \alpha - \frac{\sin \alpha \sin \beta}{\cos \beta} \right) = \cos \alpha \cos \beta - \sin \alpha \sin \beta</math>
 +
 +
<math>\fontsize{18}{27}\selectfont \tan (\alpha + \beta ) = \frac{\sin (\alpha + \beta )}{\cos (\alpha + \beta )} = \frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta - \sin \alpha \sin \beta} = \frac{\frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta}}{\frac{\cos \alpha \cos \beta - \sin \alpha \sin \beta}{\cos \alpha \cos \beta}} = \frac{\frac{\sin \alpha}{\cos \alpha} + \frac{\sin \beta}{\cos \beta}}{1 - \frac{\sin \alpha \sin \beta}{\cos \alpha \cos \beta}} = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}</math>
 +
 +
==Proof 2==
 +
 +
Let point <math>A(\cos\alpha, \sin\alpha)</math> and point <math>B(\cos\beta, \sin\beta)</math> be two points on the unit circle such that <math>\alpha > \beta</math>.
 +
 +
By the law of cosine, we know that:
 +
 +
<cmath>\cos (\alpha - \beta) = \cos \angle AOB = \frac{1^2+1^2-\vert AB \vert^2}{2 \cdot 1 \cdot 1}</cmath>
 +
 +
Apply the distance formula to obtain the length of <math>AB</math>:
 +
 +
\begin{align*}
 +
\vert AB \vert &= \sqrt{(\cos\alpha - \cos\beta)^2 + (\sin\alpha - \sin\beta)^2} \\
 +
&= \sqrt{(\cos^2\alpha + \sin^2\alpha) + (\cos^2\beta + \sin^2\beta) - 2\cos\alpha\cos\beta - 2\sin\alpha\sin\beta} \\
 +
&= \sqrt{2-2\cos\alpha\cos\beta - 2\sin\alpha\sin\beta}
 +
\end{align*}
 +
 +
Substituting and rearranging to get:
 +
 +
<cmath>\cos (\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta</cmath>
 +
 +
See that the identity holds true (and makes sense geometrically) when <math>\alpha < \beta</math> due to the fact that <math>\cos{(\beta - \alpha)} = \cos{(\alpha - \beta)}</math>.
 +
 +
Then, let <math>\theta = -\beta</math> and substitute it into the identity and the angle addition identity for cosine follows.
 +
 +
~[[User:Bloggish|Bloggish]]
  
 
==See Also==
 
==See Also==
 
* [[Trigonometric identities]]
 
* [[Trigonometric identities]]

Latest revision as of 10:05, 18 March 2025

The trigonometric angle addition identities state the following identities:

$\sin(x + y) = \sin (x) \cos (y) + \cos (x) \sin (y)$

$\cos(x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$

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

This article is a stub. Help us out by expanding it.

Proofs

Proof 1

[asy] unitsize(216); real d = 1/cos(radians(35)); real d1 = d * cos(radians(55)); real d2 = d * sin(radians(55)); pair O = (0,0); pair A = (cos(radians(20)),0); pair B = (cos(radians(20)),sin(radians(20))); pair C = (cos(radians(20)),d2); pair D = (d1,d2); draw(O--A--B--O--D--B--O--D--C--B); dot(O); dot(B); dot(A,red); dot(C,green); dot(D,blue); label("O",O,SW); label("$\alpha$",shift(dir(10)/5)*O); label("$\beta$",shift(dir(37.5)/5)*O); label("A",A,SE,red); label("B",B,E); label("C",C,NE,green); label("D",D,dir(122.5),blue); label("$\cos \alpha$",O--A,S); label("$\sin \alpha$",A--B,E); label("1",O--B,dir(302.5)); label("$\frac{\cos \alpha \sin \beta}{\cos \beta}$",B--C,E); label("$\frac{\sin \alpha \sin \beta}{\cos \beta}$",C--D,N); label("$\frac{\sin \beta}{\cos \beta}$",B--D,dir(200)); label("$\frac{1}{\cos \beta}$",D--O,dir(325)); [/asy]

$\fontsize{18}{27}\selectfont \sin (\alpha + \beta ) = \frac{\left( \sin \alpha + \frac{\cos \alpha \sin \beta}{\cos \beta} \right)}{\frac{1}{\cos \beta}} = \cos \beta \times \left( \sin \alpha + \frac{\cos \alpha \sin \beta}{\cos \beta} \right) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$

$\fontsize{18}{27}\selectfont \cos (\alpha + \beta ) = \frac{\left( \cos \alpha - \frac{\sin \alpha \sin \beta}{\cos \beta} \right)}{\frac{1}{\cos \beta}} = \cos \beta \times \left( \cos \alpha - \frac{\sin \alpha \sin \beta}{\cos \beta} \right) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$

$\fontsize{18}{27}\selectfont \tan (\alpha + \beta ) = \frac{\sin (\alpha + \beta )}{\cos (\alpha + \beta )} = \frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta - \sin \alpha \sin \beta} = \frac{\frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta}}{\frac{\cos \alpha \cos \beta - \sin \alpha \sin \beta}{\cos \alpha \cos \beta}} = \frac{\frac{\sin \alpha}{\cos \alpha} + \frac{\sin \beta}{\cos \beta}}{1 - \frac{\sin \alpha \sin \beta}{\cos \alpha \cos \beta}} = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$

Proof 2

Let point $A(\cos\alpha, \sin\alpha)$ and point $B(\cos\beta, \sin\beta)$ be two points on the unit circle such that $\alpha > \beta$.

By the law of cosine, we know that:

\[\cos (\alpha - \beta) = \cos \angle AOB = \frac{1^2+1^2-\vert AB \vert^2}{2 \cdot 1 \cdot 1}\]

Apply the distance formula to obtain the length of $AB$:

\begin{align*} \vert AB \vert &= \sqrt{(\cos\alpha - \cos\beta)^2 + (\sin\alpha - \sin\beta)^2} \\ &= \sqrt{(\cos^2\alpha + \sin^2\alpha) + (\cos^2\beta + \sin^2\beta) - 2\cos\alpha\cos\beta - 2\sin\alpha\sin\beta} \\ &= \sqrt{2-2\cos\alpha\cos\beta - 2\sin\alpha\sin\beta} \end{align*}

Substituting and rearranging to get:

\[\cos (\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta\]

See that the identity holds true (and makes sense geometrically) when $\alpha < \beta$ due to the fact that $\cos{(\beta - \alpha)} = \cos{(\alpha - \beta)}$.

Then, let $\theta = -\beta$ and substitute it into the identity and the angle addition identity for cosine follows.

~Bloggish

See Also

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