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Www.le.ac.uk Trigonometry Department of Mathematics University of Leicester.

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Presentation on theme: "Www.le.ac.uk Trigonometry Department of Mathematics University of Leicester."— Presentation transcript:

1 Trigonometry Department of Mathematics University of Leicester

2 Content Sec, Cosec and CotIntroductionInverse FunctionsTrigonometric Identities

3 Introduction – Sin, Cos and Tan Next Trigonometry is the study of triangles and the relationships between their sides and angles. These relationships are described using the functions and Sec, Cosec and Cot Intro Inverse Functions Trig Identities

4 Sine and Cosine are periodic and have the following graphs: Introduction – Sin, Cos and Tan Sine starts half way up one of the peaks. Cosine starts at the top of one of the peaks. Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

5 Introduction – Sin, Cos and Tan Next Sine and Cosine keep repeating themselves. We can use the following results to make sure we find all the solutions in a particular interval: Sec, Cosec and Cot Intro Inverse Functions Trig Identities

6 Trig Identities – Double Angle Formulae The following 2 rules hold for any values of x: Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities Click here to see a geometric proof

7 Add in these lines: Continue... a 1 b Draw these triangles: a 1 b This angle is also a a

8 From the bottom triangle, and so and Continue... a 1 b a

9 From the top-right triangle, and so and Continue... a 1 b a

10 Go back Then and a 1 b a

11 Trig Identities – To prove this, we draw this triangle: Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities By trigonometry, height =, width =, So by Pythagoras,. a 1

12 Trig Identities Using identities, we can write and in terms of : Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities (1)― (2):(1) + (2): (1) (2)

13 Trig Identities: Example 1 Write in terms of. Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

14 Trig Identities: Example 2 Write in the form Expand : We want So we want and... Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

15 Trig Identities: Example So ie. so And, so So Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

16 We get: So Write in the form Find a solution in the range to: (give your answer to 3 dp) Question... Sec, Cosec and Cot Intro Inverse Functions Trig Identities Next

17 Inverse Functions Sine, Cosine and Tangent all have inverses:, and are also called,, and. Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

18 Question... Which of the following is equivalent to ? Sec, Cosec and Cot Intro Inverse Functions Trig Identities

19 x y 0 1 Important values of sin and sin -1 : Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

20 Next Match the following: (type the letter in the box) Sec, Cosec and Cot Intro Inverse Functions Trig Identities

21 Next True or False? Sec, Cosec and Cot Intro Inverse Functions Trig Identities

22 Next Find the following: Sec, Cosec and Cot Intro Inverse Functions Trig Identities

23 Inverse Functions Next The graphs of,, can be obtained by reflecting, and in the line. (see powerpoint on Inverse Functions) However,, and are not one-to-one, so we have to use a part of the function that is one-to-one. Sec, Cosec and Cot Intro Inverse Functions Trig Identities

24 Inverse Functions We restrict to the following domains: Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

25 The inverse functions look like: Click on the graphs to see how the inverse is formed. Inverse Functions Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

26 x y = x

27 y x Go back

28 y x y = x

29 y x Go back

30 y x y = x

31 y x Go back

32 Solving Equations using Graphs Next To solve : –Let, so we’re dealing with –Find one solution using –Find another solution using –Find all the other solutions by adding and subtracting multiples of. –Find the final answer for. Use the next slide to see how this works. Sec, Cosec and Cot Intro Inverse Functions Trig Identities

33 Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

34 There are three other functions, secant, cosecant and cotangent. These are defined as: Sec, Cosec and Cot Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

35 The graphs of sec, cosec and cot are: Sec, Cosec and Cot There are asymptotes where sinx=0., so There are asymptotes where cosx=0., so There are asymptotes where tanx=0. Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

36 Sec, Cosec and Cot - Identities Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

37 Sec, Cosec and Cot – Solving Equations Example Find one solution to: –We have the identity, –Substituting this in gives –Use use the quadratic formula: Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

38 Sec, Cosec and Cot – Solving Equations – –so –so will do. Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

39 Question Find all solutions for x to this equation:, in the region. (give your answers to 3 dp, separated by commas) Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

40 Sin, Cos and Tan define the relationships between angles of a triangle. They also have inverse functions. Cosec, Sec and Cot are the recipricols of Sin, Cos and Tan. There are Trigonometric Identities which are useful for solving Trigonometric Equations. Conclusion Next Sec, Cosec and Cot Intro Inverse Functions Trig Identities

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