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the Further Mathematics network www.fmnetwork.org.uk

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the Further Mathematics network www.fmnetwork.org.uk FP2 (MEI) Calculus (part 1) Using trigonometric identities in integration, the inverse trigonometric functions, differentiation of functions involving inverse trigonometric functions. Let Maths take you Further…

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Before you start: You need to be familiar with the laws of indices (Core 1 chapter 5) and logarithms (Core 2 chapter 11). You need to have covered all of the work on functions in Core 3 chapter 3. In particular, the section on inverse trigonometrical functions on pages 45 - 46 is a useful introduction. You need to be confident with all the techniques of differentiation and integration in C2 and C3, in particular; differentiation using the chain rule, differentiation of trigonometric functions, implicit differentiation (C3 chapter 4), integration by substitution and integration of trigonometric functions (C3 chapter 5). You must also be confident with all the work on Trigonometry covered so far (C2 chapter 10 and C4 chapter 8). In particular, the enrichment work on pages 218 – 222 of the A2 Pure Mathematics textbook covers some of the work in this section. Using trigonometric identities in integration, the inverse trigonometric functions, differentiation of functions involving inverse trigonometric functions.

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When you have finished… You should: Be able to use trigonometric identities to integrate functions such as sin 2 x, sin 3 x, sin 4 x, tan x. Understand the definitions of inverse trigonometric functions. Be able to differentiate inverse trigonometric functions.

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Calculus - Reminder

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Integration of powers of sine and cosine We can use this result to integrate odd powers of sine for example:

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Try:

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Even powers of sine and cosine

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Inverse trigonometric functions It is useful to look at the graph of a function together with its inverse (use of autograph)

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arcsin

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arccos

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arctan

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Look at y=arcsecx on autograph and consider its domain and range (if time permits) Example: show that

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Differentiating inverse trigonometric functions Use autograph to draw the gradient function of y=arcsinx

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Summary of results (these are given in the exam formula book) Now that we have these results we can use the chain rule to differentiate composite functions that include inverse trigonometric functions

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Using trigonometric identities in integration, The inverse trigonometric functions, Differentiation of functions involving inverse trigonometric functions. When you have finished… You should: Be able to use trigonometric identities to integrate functions such as sin 2 x, sin 3 x, sin 4 x, tan x. Understand the definitions of inverse trigonometric functions. Be able to differentiate inverse trigonometric functions.

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Independent study: Using the MEI online resources complete the study plan for Calculus 1 Do the online multiple choice test for this and submit your answers online.

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Warm up Find the values of θ for which cot θ = 1 is true. Write the equation for a tangent function whose period is 4π, phase shift 0, and vertical shift.

Warm up Find the values of θ for which cot θ = 1 is true. Write the equation for a tangent function whose period is 4π, phase shift 0, and vertical shift.

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