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Trigonometry
Department of MathematicsUniversity of Leicester
Content
Sec, Cosec and Cot
Introduction
Inverse Functions
Trigonometric Identities
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
•
,sin xy ,cos xy .tan xy
Sec, Cosec and Cot
IntroInverse
Functions
x
xx
cos
sintan
Trig Identities
• Sine and Cosine are periodic and have the following graphs:
Introduction – Sin, Cos and Tan
xy sin xy cos
Sine starts half way up one of the
peaks.
Cosine starts at the top of one of the
peaks.
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
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:
xx
xx
xx
xx
cos)2cos(
sin)2sin(
cos)2cos(
sin)sin(
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Trig Identities – Double Angle Formulae• The following 2 rules hold for any values of
x:
abbaba cossincossin)sin(
abbaba sinsincoscos)cos(
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
aaaaaaaaa cossin2 cossincossin)sin( )2sin( ,particularIn
aaaaaaaaa 22 sincos sinsincoscos)cos( )2cos( ,particularIn
Click here to see a geometric proof
Add in these lines:
Continue...
a
1
b
Draw these triangles:
a
1
b
This angle is also a
a
From the bottom triangle, and
so and
b
xa
cossin
bax cossinb
va
coscos
bav coscosContinue...
a
1
b
a
x
y
u
v
bsin
bcos
From the top-right triangle, and
so and
b
ya
sincos
abx cossin
v
bau sinsinb
ua
sinsin
u
Continue...
a
1
b
a
x
y bsin
bcos
Go back
Then
and
v
u
abbayxba cossincossin)sin(
babauvba sinsincoscos)cos(
a
1
b
a
x
y bsin
bcos
Trig Identities –
• To prove this, we draw this triangle:
1cossin 22 xx
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
• By trigonometry, height = , width = ,
• So by Pythagoras, .1cossin 22 xx
xsin xcos
a
1
Trig Identities
• Using identities, we can write and in terms of :
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
x2cosx2sinx2cos
(1)― (2): (1) + (2):
xxx
xx
2cossincos
1sincos22
22
(1)
(2)
xx 2cos1sin2 2 xx 2cos1cos2 2
Trig Identities: Example 1
Write in terms of . )3sin( x xsin
xx
xxx
xxx
xxxxxx
xxxx
3
32
32
22
sin4sin3
sin )sin1( sin3
sincossin3
cos)cossin2()sin(cossin
cos)2sin()2cos(sin
)3sin( x
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
)2sin( Expand xx
)2sin( and )2cos( Expand xx
1cossin Use 22 xx
Trig Identities: Example 2
Write in the form
Expand :
We want
So we want and ...
xx sin3cos2
xaRaxRaxR cossincossin)sin(
)sin( axR
)sin( axR
2sin aR 3cos aR
xx cos 2 sin 3 xaRxaR cos sinsin cos
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Trig Identities: Example
So ie. so
And , so
So
,3
2
cos
sin
aR
aR ,3
2tan a 588.0a
2sin aR 6.3sin
2
aR
)588.0sin(6.3sin3cos2 xxx
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
• Find a solution in the range to:
(give your answer to 3 dp)
Question...
2sin5cos4 xx2
to0
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Next
Check answer
Show model answer
Clear answers
Hint
Inverse Functions
• Sine, Cosine and Tangent all have inverses:
, and are also called , , and .
yx 1sin xy sin yx 1cosxy cos yx 1tan xy tan
1sin 1cos 1tan arcsinarccos arctan
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Question...
Which of the following is equivalent to ? )4sin(3 32 xxy
)4(3)(sin 321 xxy
)3(sin4 213 yxx
)4(sin3 312 yyx
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
12
sin
2
3
3sin
2
1
4sin
2
1
6sin
21sin 1
62
1sin 1
42
1sin 1
32
3sin 1
x
y
0
1 xy sin
2
4
3
212123
6
Important values of sin and sin-1:
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Show
Clear
Show
Show
Show
Show
ShowShow
Show
Next
Match the following:(type the letter in the box)
6
5sin
4
3sin
2
3sin
2
1 )
2
3 )
1- )
c
b
a
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Check Answers
Clear Answers
Show Answers
Next
True or False?
4sin
4cos
3cos
3cos
2
2
4
7cos
2
1
3cos
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
True
True
True
True
False
False
False
False
Check Answers
Clear Answers
Show Answers
Next
Find the following:
2tan
3tan
32
3
23
2
1
1
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Check Answers
Clear Answers
Show Answers
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.
xy sin xy cosxy tan
xy 1sin xy 1cos xy 1tan
xy
sin cos tan
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Inverse Functions
xy sin xy cos xy tan
22
x x0
22
x
• We restrict to the following domains:
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
• The inverse functions look like:
• Click on the graphs to see how the inverse is formed.
Inverse Functions
Next
xy 1sin xy 1cos xy 1tan
2sin
21
x x1cos02
tan2
1 x
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
x
y = x
sin
y
x
y = x
1sin
sin
Go back
y
x
y = x
cos
y
x
y = x
1cos
cos
Go back
y
x
y = x
tan
y
x
y = x
1tan tan
Go back
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.
cbax )sin(baxu cu sin
cu 10 sin
2x
01 uu
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Next
sin( x )
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Make substitution
Start Again
Find one solution for u
Find other solutions for u
Find solutions for x
Find the region for u Find 2nd solution for u
• There are three other functions, secant, cosecant and cotangent. These are defined as:
Sec, Cosec and Cot
xx
sin
1sec
xx
cos
1 cosec
xx
tan
1cot
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
• The graphs of sec, cosec and cot are:
Sec, Cosec and Cot
xsec x cosec xcot
There are asymptotes where
sinx=0.
, so
There are asymptotes where
cosx=0.
, so
There are asymptotes where
tanx=0.1|sin| x 1
sin
1
x1|cos| x 1
cos
1
x
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Sec, Cosec and Cot - Identities
xx 22 coseccot1
Next
xx 22 sectan1
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Click here for a proof
Click here for a proof
(Hide)
(Hide)
Sec, Cosec and Cot – Solving Equations Example Find one solution to:
– We have the identity,
– Substituting this in gives
– Use use the quadratic formula:
5 coseccot2 xx
1coseccot 22 xx
06 coseccosec2 xx
2or 32
2411 cosec
x
Next
xx
xx22
22
coseccot1
sectan1
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Sec, Cosec and Cot – Solving Equations–
– so
– so will do.
2or 3 cosec x
2
1or
3
1sin x
6
x
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Question
• Find all solutions for x to this equation: , in the region .
• (give your answers to 3 dp, separated by commas)
08tansec6 2 xx
xx
xx22
22
coseccot1
sectan1
2 to0
Next
Sec, Cosec and Cot
IntroInverse
FunctionsTrig
Identities
Check Answers
Show Answers
• 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
IntroInverse
FunctionsTrig
Identities