E-StoreALL THE TRIGNOMETRIC FORMULAS IN ONE PLACE
ALL THE TRIGNOMETRIC FORMULAS IN ONE PLACE
Law of Cosines Law of Sines Basic Relations Laws Law of Sines Angle Sum Relations Double Angle Relations Multiple Angle Relations Half Angle Relations Production Relations Sum and Difference Relations Power Relations Adding Signals with the Same Frequency or where: Exponential Relations where: Angle addition formulas express trigonometric functions of sums of angles The first four of these are known as the prosthaphaeresis formulas, or sometimes as Simpson's formulas. The sine and cosine angle addition identities can be compactly summarized by the matrix equation These formulas can be simply derived using complex exponentials and the Euler formula as follows. Equating real and imaginary parts then gives (1) and (3), and (2) and (4) follow immediately by substituting Taking the ratio of (1) and (3) gives the tangent angle addition formula The double-angle formulas are Multiple-angle formulas are given by and can also be written using the recurrence relations The angle addition formulas can also be derived purely algebraically without the use of complex numbers. Consider the small right triangle in the figure above, which gives Now, the usual trigonometric definitions applied to the large right triangle give Solving these two equations simultaneously for the variables These can be put into the familiar forms with the aid of the trigonometric identities and which can be verified by direct multiplication. Plugging (?) into (?) and (38) into (?) then gives as before. A similar proof due to Smiley and Smiley uses the left figure above to obtain from which it follows that Similarly, from the right figure, so Similar diagrams can be used to prove the angle subtraction formulas (Smiley 1999, Smiley and Smiley). In the figure at left, giving Similarly, in the figure at right, giving A more complex diagram can be used to obtain a proof from the An interesting identity relating the sum and difference tangent formulas is given by Formulas expressing trigonometric functions of an angle The corresponding hyperbolic function double-angle formulas are Half-angle formulas and formulas expressing trigonometric functions of an angle The corresponding hyperbolic function half-angle formulas are
Triangle ABC is any triangle with side lengths a,b,c
Trigonometric Formulas
Law of Cosines 
Radius of Inscribed Circle 
Radius of Circumscribed Circle 
in terms of functions of 
and 
. The fundamental formulas of angle addition in trigonometry are given by
![[cosalpha sinalpha; -sinalpha cosalpha][cosbeta sinbeta; -sinbeta cosbeta]=[cos(alpha+beta) sin(alpha+beta); -sin(alpha+beta) cos(alpha+beta)].](http://mathworld.wolfram.com/images/equations/TrigonometricAdditionFormulas/NumberedEquation1.gif)
for 
.
![sum_(k=0)^(n)(n; k)cos^kxsin^(n-k)xsin[1/2(n-k)pi]](http://mathworld.wolfram.com/images/equations/TrigonometricAdditionFormulas/Inline65.gif)
![sum_(k=0)^(n)(n; k)cos^kxsin^(n-k)xcos[1/2(n-k)pi],](http://mathworld.wolfram.com/images/equations/TrigonometricAdditionFormulas/Inline68.gif)
![2sin[(n-1)x]cosx-sin[(n-2)x]](http://mathworld.wolfram.com/images/equations/TrigonometricAdditionFormulas/Inline71.gif)
![2cos[(n-1)x]cosx-cos[(n-2)x]](http://mathworld.wolfram.com/images/equations/TrigonometricAdditionFormulas/Inline74.gif)
![(tan[(n-1)x]+tanx)/(1-tan[(n-1)x]tanx).](http://mathworld.wolfram.com/images/equations/TrigonometricAdditionFormulas/Inline77.gif)
and 
then immediately gives
identity (Ren 1999). In the above figure, let 
. Then
in terms of functions of an angle 
,
in terms of functions of an angle 
. For real 
,
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