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== | ||
| + | |||
| + | <asy> | ||
| + | unitsize(216); | ||
| + | pair O = (0,0); | ||
| + | pair A = (cos(radians(20)),0); | ||
| + | pair B = (cos(radians(20)),sin(radians(20))); | ||
| + | pair C = (cos(radians(20)),sin(radians(55))); | ||
| + | pair D = ((cos(radians(55))*sin(radians(35))),(sin(radians(55))*sin(radians(35)))); | ||
| + | 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("$\frac{\sin \alpha \sin \beta}{\cos \beta}$",B--D,dir(); | ||
| + | </asy> | ||
==See Also== | ==See Also== | ||
* [[Trigonometric identities]] | * [[Trigonometric identities]] | ||
Revision as of 19:24, 13 January 2024
The trigonometric angle addition identities state the following identities:
This article is a stub. Help us out by expanding it.
Proofs
unitsize(216);
pair O = (0,0);
pair A = (cos(radians(20)),0);
pair B = (cos(radians(20)),sin(radians(20)));
pair C = (cos(radians(20)),sin(radians(55)));
pair D = ((cos(radians(55))*sin(radians(35))),(sin(radians(55))*sin(radians(35))));
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("$\frac{\sin \alpha \sin \beta}{\cos \beta}$",B--D,dir();
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