Trigonometry Differentiation: Derivatives of Sin and Cos
Differentiation of sine and cosine functions from first principles, step-by-step. Understand how to find the derivatives of sin(x) and cos(x) using trigonometric principles. Detailed explanations and visual aids included.
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Trig differentiation Sin x cos x
Differentiation: Trigonometry & first principles BAT prove the derivatives of sin and cos from first principles KUS objectives Starter: differentiate one of these from first principles ? = 3? ? = 4? + 4 ? = ?2
Notes Differentiation from first principles This is a lowercase delta , representing a small increase in x ? ???? ?? ? ? ???? ?? ? ???????? = ?(? + ??) ?(?) ? + ?? ? ? ? + ?? ?(?) ?? ???????? = (x+ x, f(x+ x)) ???????? = (x, f(x)) ? ? + ?? ?(?) ?? ? (?) = lim ?? 0 Gradient = change in y change in x This is the definition for differentiating a function f(x+ x) f(x) We use h for ?? x+ x - x
y = x2 Example differentiation of y = x2 From first principles ? ? + ? ?(?) ?? ? (?) = lim ?? 0 (? + )2 ?2 ? + ?? ? Multiply the bracket ???????? = lim ?? 0 ?2+ 2( )(?) + ( )2 ?2 ? + ? Group some terms ???????? = lim ?? 0 2 ? + 2 Cancel h ???????? = lim ?? 0 At the original point, h = 0 ???????? = lim ?? 02? + ? (?) = ??
WB15a Derivative of sin x part 1 First principles ? ???(?) = ? ? + ? ? ? ? ??? ? 0 By the addition rule for sin(? + ?) ??? ? + ??? ? ??? 0 sin?cos + cos? sin sin? ??? 0 = Split up and rearrange sin? cos 1 ??? 0 cos? sin ??? 0 = + cos 1 ??? 0 sin ??? 0 = sin? + cos?
WB15b Derivative of sin x part 2 ? ?? cos 1 ??? 0 sin ??? 0 + cos? sin? = sin? As h 0 cos h 1 ??? 0 cos 1 = 0 = cos 1 0 0 As h 0 sin ??? 0 = = 1 ? = and sin approach each other and ? ??sin? = sin? ? + cos? ? sin 1 1 ? ?????? = ????
16a Derivative of cos x part 1 First principles ? ???(?) = ? ? + ? ? ? ? ??? ? 0 By the addition rule for cos(? + ?) ??? ? + ??? ? ??? 0 cos?cos sin? sin cos? ??? 0 = Split up and rearrange cos? cos 1 ??? 0 sin? sin ??? 0 = + cos 1 ??? 0 sin ??? 0 = cos? sin?
WB16b Derivative of cos x part 2 ? ?? cos 1 sin ??? 0 ??? 0 sin? cos? = cos? As h 0 cos h 1 ??? 0 cos 1 = 0 = cos 1 0 0 As h 0 sin ??? 0 = = 1 ? = and sin approach each other and ? ??cos? = cos? ? sin? ? sin ? ?????? = ???? 1 1
KUS objectives BAT prove the derivatives of sin and cos from first principles self-assess One thing learned is One thing to improve is
? ? + ?? ?(?) ?? You need to be able to differentiate Trigonometric Functions A ? (?) = lim ?? 0 Let f(x) = sinx (Angle x is in radians) ??? ? + ?? ???(?) ?? ? (?) = lim r ?? 0 Multiply the function using sin(A + B) x B r ????????? + ????????? ???? ?? O ? (?) = lim ?? 0 As x 0 Cos x 1 Sin x x ????(1) + ????(??) ???? ?? ? (?) = lim 1 2?2? ?? 0 Area of the sector OAB: Simplify terms ???? + ?????? ???? ?? 1 2?2???? Area of the triangle OAB: ? (?) = lim ?? 0 Sinx s cancel out As x approaches 0, the area of the triangle and sector become equal. Hence: 1 2?2???? =1 ?????? ?? ? (?) = lim ?? 0 2?2? Cancel x s ? ? = ???? ???? = ?
You need to be able to differentiate Trigonometric Functions Gradient = -1 y = Cos If: ? = ???? 1 y = Sin ?? ??= ???? Then: 0 /2 3 /2 2 -1 Gradient = 0 If: ? = ???? ? ?? ??= ? (?)????(?) This means that the Cos graph is actually telling you the gradient of the Sin graph at the equivalent point! Then: At , Cos = -1 The gradient of Sin at is -1 At 3 /2, Cos = 0 The gradient of Sin at 3 /2 is 0!
? ? + ?? ?(?) ?? You need to be able to differentiate Trigonometric Functions ? (?) = lim ?? 0 Let f(x) = cosx ??? ? + ?? ???(?) ?? ? (?) = lim ?? 0 Multiply the function using cos(A + B) ????????? ????????? ???? ?? ? (?) = lim ?? 0 As x 0 Cos x 1 Sin x x ???? 1 ????(??) ???? ?? ? (?) = lim ?? 0 Simplify terms ???? ?????? ???? ?? ? (?) = lim ?? 0 Cosx s cancel out ?????? ?? ? (?) = lim ?? 0 Cancel x s ? ? = ????
