# Trigonometry

```PROGRAMME F8
TRIGONOMETRY
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Worked examples and exercises are in the text
Programme F8: Trigonometry
Angles
Trigonometric identities
Trigonometric formulas
NB: I have slightly edited the book’s slides
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Angles
Rotation
Triangles
Trigonometric ratios
Reciprocal ratios
Pythagoras’ theorem
Special triangles
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Worked examples and exercises are in the text
Programme F8: Trigonometry
Angles
Rotation
When a straight line is rotated about a point it sweeps out an angle that can
be measured in degrees or radians
A straight line rotating through a full angle and returning to its starting point
is said to have rotated through 360 degrees (360o )
One degree = 60 minutes (60'), and one minute = 60 seconds (60'')
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Angles
When a straight line of length r is rotated about one end so that the other
end describes an arc of length r the line is said to have rotated through 1
STROUD
Worked examples and exercises are in the text
All at Sea
(added by John Barnden)
The circumference of the Earth is about 24,900 miles.
That corresponds to 360 x 60 minutes of arc, = 21,600'
So 1' takes you about 24,900/21,600 miles = about 1.15 miles.
A nautical mile was originally defined as being the distance that one minute
of arc takes you on any meridian (= line of longitude). This distance varies a
bit as you go along the meridian, because of the irregular shape of the Earth.
A nautical mile is now defined as 1852 metres, which is about 1.15 miles.
A knot is one nautical mile per hour. NB: 60 knots is nearly 70 miles/hour.
Look up nautical miles and knots on the web – it’s interesting.
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Angles
Triangles
All triangles possess
shape and size. The
shape of a triangle is
governed by the three
angles and the size by
the lengths of the three
sides
AB
AC
BC


AB AC BC
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Angles
Trigonometric ratios
AB
AC
BC


AB AC  BC 
so that:
AB AB
AB AB
AC AC 

and

and

AC AC 
BC BC 
BC BC 
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Trigonometric ratios:
in a right-angled triangle
AB = the “hypotenuse”
sine of angle  
AC
- denoted by sin
AB
cosine of angle  
BC
- denoted by cos
AB
tangent of angle  
BC
- denoted by tan
AB
Error in tangent formula: should be AC/BC
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Reciprocal ratios
cosecant of angle  
secant of angle  
1
- denoted by cos ec
sin 
1
- denoted by sec
cos
cotangent of angle  
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1
- denoted by cot 
tan 
Worked examples and exercises are in the text
Programme F8: Trigonometry
Pythagoras’ s Theorem
The square on the hypotenuse of a rightangled triangle is equal to the sum of the
squares on the other two sides
a 2  b2  c2
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Special triangles
Right-angled isosceles
Angles measured in degrees:
sin 45  cos 45 
1
and tan 45  1
2
Angles measured in radians:
1
sin  / 4  cos  / 4 
and tan  / 4  1
2
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Special triangles, contd
Half equilateral
Angles measured in degrees:
sin 30  cos 60 
sin 60  cos30 
tan 60 
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1
2
3
2
1
 3
tan 30
Worked examples and exercises are in the text
Programme F8: Trigonometry
Angles
Special triangles
Half equilateral
Angles measured in radians:
sin  / 6  cos  / 3 
sin  / 3  cos  / 6 
tan  / 3 
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1
tan  / 6
1
2
3
2
 3
Worked examples and exercises are in the text
Programme F8: Trigonometry
Angles
Trigonometric identities
Trigonometric formulas
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Worked examples and exercises are in the text
Programme F8: Trigonometry
Trigonometric identities
The fundamental identity
Two more identities
Identities for compound angles
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Worked examples and exercises are in the text
Programme F8: Trigonometry
The fundamental identity
The fundamental trigonometric identity
is derived from Pythagoras’ theorem
a b c
2
2
2
so
a 2 b2
 2 1
2
c
c
that is:
cos 2   sin 2   1
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Two more identities
Dividing the fundamental identity by cos2
cos   sin   1
2
2
so that
cos 2  sin 2 
1


cos 2  cos 2  cos 2 
that is:
1  tan 2   sec 2 
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Last one:
Dividing the fundamental identity by sin2
cos   sin   1
2
2
so that
cos 2  sin 2 
1


sin 2  sin 2  sin 2 
that is:
cot 2   1  cosec 2 
STROUD
Worked examples and exercises are in the text
Beyond Pythagoras
Ignore the outer triangle.
(added by John Barnden)
Let the sides of the inner triangle
ABC have lengths a, b, c (opposite
the angles A, B, C, respectively).
Then:
c2 = a2 + b2 – 2ab.cos C
This works for any shape of triangle.
When C = 90 degrees, we just get
Pythagoras, as cos 90o = 0.
EX: What happens when C is zero?
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Worked examples and exercises are in the text
Beyond Pythagoras, contd
The result on the previous slide can
easily be shown be dropping a
perpendicular from vertex A to line
BC. Try it as an EXERCISE.
Use Pythagoras on each of the
resulting right-angle triangles.
You’ll also need to use the
Fundamental Identity.
STROUD
Worked examples and exercises are in the text
Another Interesting Fact
(added by John Barnden)
a/sin A = b/sin B = c/sin C
This again can easily be seen by
dropping a perpendicular from any
vertex to the opposite side. Try it as
an EXERCISE.
Just use the definition of sine twice to
get two different expressions for the
length of the perpendicular.
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Worked examples and exercises are in the text
Programme F10: Functions
Switiching to Programme F10 briefly –
reminder of part of Bohnet’s coverage in Term 1
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Worked examples and exercises are in the text
Programme F10: Functions
Trigonometric functions
Rotation
For angles greater than zero and less than /2 radians the trigonometric
ratios are well defined and can be related to the rotation of the radius of a
unit circle:
STROUD
Worked examples and exercises are in the text
Programme F10: Functions
Trigonometric functions
Rotation
By continuing to rotate the radius of a unit circle the trigonometric ratios
can extended into the trigonometric functions, valid for any angle:
STROUD
Worked examples and exercises are in the text
Programme F10: Functions
Trigonometric functions
Rotation
The sine function:
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Worked examples and exercises are in the text
Programme F10: Functions
Trigonometric functions
Rotation
The cosine function:
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Worked examples and exercises are in the text
Programme F10: Functions
Trigonometric functions
The tangent
The tangent is the ratio of the sine to the cosine:
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tan 
sin 
cos
Worked examples and exercises are in the text
Programme F8: Trigonometry
Switching back to Programme F8
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Angles
Trigonometric identities
Trigonometric formulas
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Trigonometric formulas
Sums and differences of angles
Double angles
Sums and differences of ratios
Products of ratios
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Trigonometric formulas
Sums and differences of angles
(NB: there’s a typo on LHS of 2nd sine
formula – John B.)
cos(   )  cos cos   sin  sin 
cos(   )  cos cos   sin  sin 
sin(   )  sin  cos   cos sin 
sin(   )  sin  cos   cos sin 
tan   tan 
1  tan  tan 
tan   tan 
tan(   ) 
1  tan  tan 
tan(   ) 
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Double angles
cos 2  cos 2   sin 2 
cos 2  1  2sin 2 
cos 2  2cos 2   1
sin 2  2sin  cos 
tan 2 
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2 tan 
1  tan 2 
Worked examples and exercises are in the text
Programme F8: Trigonometry
Sums and differences of trig functions
sin   sin   2sin
 
cos
 
2
2
   
sin   sin   2cos
sin
2
2
cos  cos   2cos
 
cos
 
2
2
   
cos  cos   2sin
sin
2
2
STROUD
Worked examples and exercises are in the text
Programme F8: Trigonometry
Products of trig functions
2sin  cos   sin(   )  sin(   )
2cos cos   cos(   )  cos(   )
2sin  sin   cos(   )  cos(   )
STROUD
Worked examples and exercises are in the text
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