1. Department of Mathematics University of Leicester
Trigonometry
Department of Mathematics
University of Leicester

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

3. 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
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4. 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.
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5. 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
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6. 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
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7. 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...

8. From the bottom triangle, and so and1 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...

9. From the top-right triangle, and so and1 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...

10. 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

11. 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,
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12. 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):
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13. Trig Identities: Example 1
Intro
Trig Identities
Inverse Functions
Sec, Cosec and Cot
Trig Identities: Example 1
Write in terms of
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14. 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.
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15. 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.
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16. 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
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17. 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
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18. 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

19. 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
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20. 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)
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21. True or False? Intro Trig Identities Inverse Functions
Sec, Cosec and Cot
True or False?
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22. Find the following: Intro Trig Identities Inverse Functions
Sec, Cosec and Cot
Find the following:
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23. 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.
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24. 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:
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25. 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.
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26. y = x x

27. y
y = x x
Go back

28. y
y = x x

29. y
y = x x
Go back

30. y
y = x x

31. y
y = x x
Go back

32. 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
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33. 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
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34. 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
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35. 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.
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36. Sec, Cosec and Cot - Identities
Intro
Trig Identities
Inverse Functions
Sec, Cosec and Cot
Sec, Cosec and Cot - Identities
(b) Is correct
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37. 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:
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38. 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.
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39. 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)
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40. 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
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