# Department of Mathematics University of Leicester

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Department of Mathematics University of Leicester
Trigonometry Department of Mathematics University of Leicester

Trigonometric Identities
Content Introduction Trigonometric Identities Inverse Functions Sec, Cosec and Cot

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

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

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

Trig Identities – Double Angle Formulae
Intro Trig Identities Inverse Functions Sec, Cosec and Cot Trig Identities – Double Angle Formulae The following 2 rules hold for any values of x: The main thing is though, is can we add in a geometric proof of cos(a+b), sin(a+b) etc. Click here to see a geometric proof Next

Draw these triangles: Add in these lines: 1 b a 1 b a Continue...
This angle is also a Add in these lines: The main thing is though, is can we add in a geometric proof of cos(a+b), sin(a+b) etc. Continue...

From the bottom triangle, and so and
1 b The main thing is though, is can we add in a geometric proof of cos(a+b), sin(a+b) etc. From the bottom triangle, and so and Continue...

From the top-right triangle, and so and
1 b The main thing is though, is can we add in a geometric proof of cos(a+b), sin(a+b) etc. From the top-right triangle, and so and Continue...

a 1 b The main thing is though, is can we add in a geometric proof of cos(a+b), sin(a+b) etc. Then and Go back

Trig Identities – To prove this, we draw this triangle:
Intro Trig Identities Inverse Functions Sec, Cosec and Cot Trig Identities – To prove this, we draw this triangle: a 1 The main thing is though, is can we add in a geometric proof of cos(a+b), sin(a+b) etc. By trigonometry, height = , width = , So by Pythagoras, Next

Trig Identities Using identities, we can write and in terms of : (1)
Intro Trig Identities Inverse Functions Sec, Cosec and Cot Trig Identities Using identities, we can write and in terms of : (1) (2) The main thing is though, is can we add in a geometric proof of cos(a+b), sin(a+b) etc. ― (2): (1) + (2): Next

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

Trig Identities: Example 2
Intro Trig Identities Inverse Functions Sec, Cosec and Cot Trig Identities: Example 2 Write in the form Expand : We want So we want and ... sin^2x + cos^2x = 1. double angle formulas. for example Write 2cos(x) + 3sin(x) in the form Rsin(x+a) which they can then use to find say 2cos(x) + 3sin(x) = 2. Hope this makes sense. If not, just let me know. Next

Trig Identities: Example
Intro Trig Identities Inverse Functions Sec, Cosec and Cot Trig Identities: Example So ie. so And , so So sin^2x + cos^2x = 1. double angle formulas. for example Write 2cos(x) + 3sin(x) in the form Rsin(x+a) which they can then use to find say 2cos(x) + 3sin(x) = 2. Hope this makes sense. If not, just let me know. Next

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

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

Question... Which of the following is equivalent to ? Intro
Trig Identities Inverse Functions Sec, Cosec and Cot Question... Which of the following is equivalent to ? (b) Is correct

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

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

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

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

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

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

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

y = x x

y y = x x Go back

y y = x x

y y = x x Go back

y y = x x

y y = x x Go back

Solving Equations using Graphs
Intro Trig Identities Inverse Functions Sec, Cosec and Cot Solving Equations using Graphs 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. (b) Is correct Next

Intro Trig Identities Inverse Functions Sec, Cosec and Cot Next
objects: 1 is the graph 2 is u0 3 is u1(=pi-u0) 4 is u1(=pi-u0)=u0 5 is the working for the big thing 6 is the answer for the big thing Next

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

Sec, Cosec and Cot The graphs of sec, cosec and cot are: Intro
Trig Identities Inverse Functions Sec, Cosec and Cot Sec, Cosec and Cot The graphs of sec, cosec and cot are: (b) Is correct 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 - Identities
Intro Trig Identities Inverse Functions Sec, Cosec and Cot Sec, Cosec and Cot - Identities (b) Is correct Next

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

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

Intro Trig Identities Inverse Functions Sec, Cosec and Cot Conclusion 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. (b) Is correct Next

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