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> | ||
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<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> | ||
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<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== | ||
| + | |||
| + | <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> | ||
==See Also== | ==See Also== | ||
* [[Trigonometric identities]] | * [[Trigonometric identities]] | ||
Latest revision as of 19:46, 13 January 2024
The trigonometric angle addition identities state the following identities:
This article is a stub. Help us out by expanding it.
Proofs
![[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]](http://latex.artofproblemsolving.com/a/7/e/a7edae3e9c92e3097608e3f5578b1e4ec29f5edc.png)
