Trigonometric Identities
that cos( B) = cosB(cos is even) and sin( B) = sinB(sin is odd) Similarly (7) comes from (6) (8) is obtained by dividing (6) by (4) and dividing top and bottom by cosAcosB, while (9) is obtained by dividing (7) by (5) and dividing top and bottom by cosAcosB (10), (11), and (12) are special cases of (4), (6), and (8) obtained by putting A= B
Trigonometric Identities Revision : 1
sin(A−B)+sin(A+B) = 2sinAcosB and dividing both sides by 2 we obtain the identity sinAcosB = 1 2 sin(A−B)+ 1 2 sin(A+B) (9) In the same way we can add equations (3) and (8) cos(A−B) = cosAcosB +sinAsinB +(cos(A+B) = cosAcosB −sinAsinB) to get cos(A−B)+cos(A+B) = 2cosAcosB which can be rearranged to yield the identity cosAcosB = 1 2
TRIGONOMETRY LAWS AND IDENTITIES
sin(A) a = sin(B) b = sin(C) c DOUBLE-ANGLE IDENTITIES sin(2x)=2sin(x)cos(x) cos(2x) = cos2(x)sin2(x) = 2cos2(x)1 =12sin2(x) tan(2x)= 2tan(x) 1 2tan (x) HALF-ANGLE IDENTITIES sin ⇣x 2 ⌘ = ± r 1cos(x) 2 cos ⇣x 2 ⌘ = ± r 1+cos(x) 2 tan ⇣x 2 ⌘ = ± s 1cos(x) 1+cos(x) PRODUCT TO SUM IDENTITIES sin(x)sin(y)= 1 2 [cos(xy)cos(x+y)] cos(x
Trigonometric Identities - Miami
= sin 1+cos Double-Angle Formulas sin2 = 2sin cos cos2 = cos2 sin2 tan2 = 2tan 1 tan2 cos2 = 2cos2 1 cos2 = 1 2sin2 Product-to-Sum Formulas sinxsiny= 1 2
Formulas from Trigonometry
(A+B)cos 1 2 (A B) cosA cosB= 2sin 1 2 (A+B)sin 1 2 (B A) sinAsinB= 1 2 fcos(A B) cos(A+B)g cosAcosB= 1 2 fcos(A B)+cos(A+B)g sinAcosB= 1 2 fsin(A B)+sin(A+B)g cos( ) = sin( +ˇ=2) Di erentiation Formulas: d dx (uv) = udv dx + du dx v d dx u v = v (du=dx )udv=dx v2 Chain rule: dy dx = dy du du dx d dx sinu= cosudu dx d dx cos u= sin du dx d dx
An elementary proof of two formulas in trigonometry
Hence sin(a+b)=AE= DE+AD=sin(a)cos(b)+cos(a)sin(b) B For general a and b, we can use that , cos(-x)=cos(x), , and etc to reduce them to the above cases We can see that the two equations are also right cos(a+b)=cos(a)cos(b)-sin(a)sin(b) sin(a+b)=sin(a)cos(b)+cos(a)sin(b) C By replacing b as –b, we have:
Euler’s Formula and Trigonometry
cos(ax)cos(bx)dx; Z cos(ax)sin(bx)dx or Z sin(ax)sin(bx)dx are usually done by using the addition formulas for the cosine and sine functions They could equally well be be done using exponentials, for instance (assuming a6= b) Z cos(ax)cos(bx)dx= Z 1 2 (eiax+ e iax) 1 2 (eibx+ e ibx)dx = 1 4 Z (ei(a+b)x+ ei(a b)x+ e i(a b)x+ e i(a+b)x)dx = 1 2
Complex numbers and Trigonometric Identities
• Distributing exponents was his only sin • But that’s enough to do an algebra student in • An example, his demise should serve, • For other students who haven’t heard, • Distributing exponents is a sin • It’s enough to do an algebra student in • • Donald E Brook • Mt San Antonio College • ????????+ ????????
Trignometrical Formulae Standard Integrals
2cosA sinB = sin(A+B)−sin(A−B) 2cosA cosB = cos(A+B)+cos(A−B) 2sinA sinB = cos(A−B)−cos(A+B) Hyperbolic Functions sinhx = ex −e−x 2, coshx = ex +e−x 2 Standard Derivatives f(x) f0(x) x nnx −1 sinax acosax cosax −asinax tanax asec2 ax e axae lnx 1 x sinhax acoshax coshax asinhax uv u0 v +uv0 u v u0 v −uv0 v2 Standard
[PDF] cos(a+b) démonstration
[PDF] sin a+sin b
[PDF] sin 2a
[PDF] tan = cos/sin ou sin/cos
[PDF] tan = sin/cos
[PDF] cos x pi 2
[PDF] cos x 2
[PDF] sin x pi 2
[PDF] cos x 1
[PDF] cos(a+b) démonstration exponentielle
[PDF] cos a+b démonstration géométrique
[PDF] tan 2a démonstration
[PDF] cos(a-b) démonstration
[PDF] cos(a+b)=cos(a) cos(b)-sin(a) sin(b) démonstration