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7: Differentiating some Trig Functions © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

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Differentiating some Trig Functions A reminder of the rules for differentiation developed so far! The chain rule ( for functions of a function ): where ( I call these functions the simple ones. )

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Differentiating some Trig Functions The trig functions are quite different in shape from either of the simple functions we’ve met so far, so the gradient functions won’t follow the same rules. We’ll start with and use degrees We only need the 1 st quadrant as symmetry will then give us the rest of the gradient function.

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Differentiating some Trig Functions x x x x The gradient function x x x x The function

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Differentiating some Trig Functions x x x x The gradient function x x x The gradient drops more slowly between and... than between and. The function x

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Differentiating some Trig Functions x x x x x x x x The gradient function The function

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Differentiating some Trig Functions x x x x x x x x The gradient function The function

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Differentiating some Trig Functions x x x x The function We can estimate the gradient at x = 0 by using the tangent. So, the gradient is The gradient function looks like BUT we need a scale on the axis.

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Differentiating some Trig Functions x x x x The function The gradient function isn’t since not. So,

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Differentiating some Trig Functions x However, if we use radians: x x x x It can be shown that this length... is exactly 1.

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Differentiating some Trig Functions From now on we will assume that x is in radians unless we are told otherwise. We have, Exercise Using radians sketch for. Underneath the sketch, sketch the gradient function. Suggest an equation for the gradient graph.

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Differentiating some Trig Functions Solution: This is a reflection of in the x -axis so its equation is

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Differentiating some Trig Functions SUMMARY If x is in radians, We need a bit more theory before we can differentiate the trig function. This is done in a later presentation. N.B. We have not proved these results; just shown they look correct.

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Differentiating some Trig Functions Compound Trig Functions We can use the chain rule to differentiate trig functions of a function. Solution: (a) First find the gradient function: e.g. 1 Find the gradient of at the point where. Let N.B. We don’t need to put brackets round 3x. N.B. Radians!

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Differentiating some Trig Functions e.g. 2 Differentiate What would you let u equal in this example? If we write as we can easily see that the inner function is. So, let Solution:

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Differentiating some Trig Functions Exercise Differentiate the following with respect to x : 1.2. Solutions: 1. Let

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Differentiating some Trig Functions 2. Let 3.

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Differentiating some Trig Functions

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The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

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Differentiating some Trig Functions A reminder of the rules for differentiation developed so far! The chain rule ( for functions of a function ): where ( I call these functions the simple ones. )

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Differentiating some Trig Functions SUMMARY If x is in radians,

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Differentiating some Trig Functions Compound Trig Functions We can use the chain rule to differentiate trig functions of a function. Solution: (a) First find the gradient function: e.g. 1 Find the gradient of at the point where. Let N.B. We don’t need to put brackets round 3x. N.B. Radians!

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Differentiating some Trig Functions e.g. 2 Differentiate If we write as we can easily see that the inner function is. So, let Solution:

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