Differentiation Trigonometry first principles KUS objectives BAT prove

Differentiation: Trigonometry & first principles • KUS objectives BAT prove the derivatives of sin and cos from first principles Starter: differentiate one of these from first principles

Notes Differentiation from first principles This is a lowercase ‘delta’, representing a small increase in x (x+δx, f(x+δx)) (x, f(x)) Gradient = change in y ÷ change in x f(x+δx) – f(x) x+δx - x

y = x 2 Example differentiation of y = x 2 From first principles Multiply the bracket Group some terms Cancel h At the original point, h=0

WB 15 a Derivative of sin x part 1 First principles Split up and rearrange

WB 15 b Derivative of sin x part 2 As h → 0 cos h → 1

16 a Derivative of cos x part 1 First principles Split up and rearrange

WB 16 b Derivative of cos x part 2 As h → 0 cos h → 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 –

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You need to be able to differentiate Trigonometric Functions A Let f(x) = sinx (Angle x is in radians) r x O r Multiply the function using sin(A + B) B As δx 0 Cosδx 1 Sinδx δx Area of the sector OAB: Simplify terms Area of the triangle OAB: Sinx’s cancel out As x approaches 0, the area of the triangle and sector become equal. Hence: Cancel δx’s

You need to be able to differentiate Trigonometric Functions If: Then: Gradient = -1 y = Cosθ 1 y = Sinθ 0 π/ 2 π 3π/ 2 -1 2π Gradient = 0 This means that the Cos graph is actually telling you the gradient of the Sin graph at the equivalent point! At π, Cosθ = -1 The gradient of Sinθ at π is -1 At 3π/2, Cosθ = 0 The gradient of Sinθ at 3π/2 is 0!

You need to be able to differentiate Trigonometric Functions Let f(x) = cosx Multiply the function using cos(A + B) As δx 0 Cosδx 1 Sinδx δx Simplify terms Cosx’s cancel out Cancel δx’s