Trigonometric Identities
A+ B 2 cos A B 2 (13) cosA cosB= 2sin A+ B 2 sin A B 2 (14) sinA+ sinB= 2sin A+ B 2 cos A B 2 (15) sinA sinB= 2cos A+ B 2 sin A B 2 (16) Note that (13) and (14) come from (4) and (5) (to get (13), use (4) to expand cosA= cos(A+ B 2 + 2) and (5) to expand cosB= cos(A+B 2 2), and add the results) Similarly (15) and (16) come from (6) and (7)
Trigonometric Identities Revision : 1
cos(A−B)+cos(A+B) = 2cosAcosB which can be rearranged to yield the identity cosAcosB = 1 2 cos(A−B)+ 1 2 cos(A+B) (10) Suppose we wanted an identity involving sinAsinB We can find one by slightly modi-fying the last thing we did Rather than adding equations (3) and (8), all we need to do is subtract equation (3) from equation (8): cos(A
Formulas from Trigonometry
Formulas from Trigonometry: sin 2A+cos A= 1 sin(A B) = sinAcosB cosAsinB cos(A B) = cosAcosB tansinAsinB tan(A B) = A tanB 1 tanAtanB sin2A= 2sinAcosA cos2A= cos2 A sin2 A tan2A= 2tanA
Trigonometric Identities - Miami
cos(x+ y) = cosxcosy sinxsiny cos(x y) = cosxcosy+ sinxsiny tan(x+ y) = tanx+tany 1 tanxtany tan(x y) = tanx tany 1+tanxtany Half-Angle Formulas sin 2 = q 1 cos 2 cos 2 = q 1+cos 2 tan 2 = q 1+cos tan 2 = 1 cosx sinx tan 2 = sin 1+cos Double-Angle Formulas sin2 = 2sin cos cos2 = cos2 sin2 tan2 = 2tan 1 tan2 cos2 = 2cos2 1 cos2 = 1 2sin2 Product
Complex numbers and Trigonometric Identities
• a(b + c) = ab + ac • But why is this even true to begin with? • Here is a visual proof where we can think of the real number values representing the lengths of rectangles and their products the area of their associated rectangles • Even the proof for natural numbers takes effort
Euler’s Formula and Trigonometry
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 Z (cos((a+ b)x) + cos((a b)x))dx = 1 2 (1 a+ b sin((a+ b)x) + 1
Formule trigonometrice a b a b c b a c - Math
; cos = b c; tg = a b; ctg = b a; (a; b- catetele, c- ipotenuza triunghiului dreptunghic, - unghiul, opus catetei a) 2 tg = sin cos ; ctg = cos sin : 3 tg ctg = 1: 4 sin ˇ 2 = cos ; sin(ˇ ) = sin : 5 cos ˇ 2 = sin ; cos(ˇ ) = cos : 6 tg ˇ 2 = ctg ; ctg ˇ 2 = tg : 7 sec ˇ 2 = cosec ; cosec ˇ 2 = sec : 8 sin 2 + cos = 1: 9 1 + tg
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
Scalar Product: ab = ab cos q a b
The equation (x – 2a) 2+ (y – b)2 = r represents a circle centre (a, b) and radius r Scalar Product: a b = ab cos q, where q is the angle between a and b a b = a 1 b 1 + aor 2 b 2 + a 3 b 3 where a = Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B cos (A ± B) = cos A cos B ± sin A sin B sin 2A = 2sin A cos A 2cos
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