trigonometry
(a)sinx ± (b)cosx or (a)cosx ± (b)sinx can be written as
Rsin(x ± α) or Rcos(x ± α)
all variations of trig identities when dividing by cos^2
Tan² + 1 = sec² 1 = sec² - Tan² Tan² = sec² - 1
proof and identity of cos(A) (half angle formula)
consider cos(2A)= cos²A - sin²A = 1-2sin²A = 2cos²A-1 replace 2A = A with A = 1/2A cos(A) = cos²(A/2) - sin²(A/2) = 1 - 2sin²(A/2) = 2cos²(A/2) - 1
proof and identities of cos(2A)
consider cos(A+B)= cosAcosB-sinAsinB let B=A: cos(2A) = cos²A-sin²A 1-2sin²A (let cos² = 1 - sin²) 2cos²A-1 (let sin² = 1 - cos²)
proof and identity of sin(A) (half angle formula)
consider sin(2A) = 2sinAcosA replace 2A = A with A = 1/2A sin(A) = 2sin(A/2)cos(A/2)
proof and identity of sin(2A)
consider sin(A+B) = sinAcosB + sinBcosA let B=A sin (A+A) = sinAcosA + sinAcosA = 2sinAcosA
proof and identity of tan(A) (half angle formula)
consider tan(2A) = (2tanA)/(1-tan^2A) replace 2A = A with A = 1/2A tan(A) = (2tanA/2)/(1-tan²A/2)
proof and identity of tan(2A)
consider tan(A+B)= (tanA + tanB)/(1 - tanAtanB) let B=A tan(2A)= (2tanA)/(1 - tan²A)
how do we derive tan (a+b) and tan(A-B)
convert into sin(A+A)/cos(A+B) convert this using the sin and cos addition formulae divide everything by the bottom left (cosAcosB) cancel down
cos(3A) =
cos(2A + A) cos2AcosA - sin2AsinA cosA(2cos²(A) - 1) - sinA(2sinAcosA) 2cos³(A) - cosA - 2sin²(A)cosA 2cos³(A) - cosA - 2cosA(1 - cos²(A)) 2cos³(A) - cosA - 2cosA + 2cos³(A) 4cos³(A) - 3cosA
cos(A+B)
cosAcosB - sinAsinB
cos(A-B)
cosAcosB+sinAsinB
when finding the min of sin or cos
set equal to -1
when finding the max of sin or cos
set equal to 1
sin(3A)=
sin(2A + A) sin2AcosA + sinAcos2A cosA(2sinAcosA) + sinA(1-2sin²(A)) 2sinAcos²(A) + sinA-2sin³(A) 2sinA(1-sin²(A)) + sinA-2sin³(A) 2sinA - 2sin³(A) + sinA - 2sin³(A) 3sinA - 4sin³(A)
sin(A+B)=
sinAcosB+cosAsinB
sin(A-B)
sinAcosB-cosAsinB
what does R =
√(a² + b²)
what is tan 60
√3
tan(A+B)
(tanA + tanB)/(1 - tanAtanB)
tan(A-B)
(tanA - tanB) / (1 + tanAtanB)
what is cos 45
(√2)/2
what is sin 45
(√2)/2
what is cos 30
(√3)/2
what is sin 60
(√3)/2
what is tan 30
(√3)/3
what is tan 45
1
all variations of trig identities when dividing by sin²x
1 + cot² = cosec² cosec²- cot² = 1 cot² = cosec² -1
what is cos 60
1/2
what is sin 30
1/2
identity for cos²(A) rearrangement of cos(2A) influential in c4 integration
1/2 + 1/2(cos2A)
identity for sin²(A) rearrangement of cos(2A) influential in c4 integration
1/2 - 1/2cos(2A)
sec x
1/cos x
cosec x
1/sin x
cot x
1/tan x
