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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(

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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