STROUD Worked examples and exercises are in the text PROGRAMME F8 TRIGONOMETRY

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Worked examples and exercises are in the text

PROGRAMME F8

TRIGONOMETRY

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Programme F8: Trigonometry

Angles

Trigonometric identities

Trigonometric formulas

NB: I have slightly edited the book’s slides

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Angles

Rotation

Radians

Triangles

Trigonometric ratios

Reciprocal ratios

Pythagoras’ theorem

Special triangles

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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'')

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Angles

Radians

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 radian – 1 rad

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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.

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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

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Angles

Trigonometric ratios

so that:

and and

AB AC BC

A B A C B C

AB A B AB A B AC A C

AC A C BC B C BC B C

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Trigonometric ratios: in a right-angled triangle

AB = the “hypotenuse”

of angle - denoted by sin

of angle - denoted by cos

of angle - denoted by tan

ACsine

AB

BCcosine

AB

BCtangent

AB

Error in tangent formula: should be AC/BC

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Reciprocal ratios

1 of angle - denoted by cosec

sin

1 of angle - denoted by sec

cos

1 of angle - denoted by cot

tan

cosecant

secant

cotangent

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Pythagoras’ s Theorem

The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides

2 2 2a b c

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Special triangles

Right-angled isosceles

Angles measured in degrees:

1sin 45 cos45 and tan 45 1

2

Angles measured in radians:

1sin / 4 cos / 4 and tan / 4 1

2

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Special triangles, contd

Half equilateral

Angles measured in degrees:

1sin30 cos60

2

3sin 60 cos30

21

tan 60 3tan30

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Angles

Special triangles

Half equilateral

Angles measured in radians:

1sin / 6 cos /3

2

3sin /3 cos / 6

21

tan /3 3tan / 6

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Angles



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The fundamental identity

Two more identities

Identities for compound angles

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The fundamental identity

The fundamental trigonometric identity is derived from Pythagoras’ theorem

2 22 2 2

2 2

2 2

so 1

that is:

cos sin 1

a ba b c

c c

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Two more identities

Dividing the fundamental identity by cos2

2 22 2

2 2 2

2 2

cos sin 1cos sin 1 so that

cos cos costhat is:

1 tan sec

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Last one:

Dividing the fundamental identity by sin2

2 22 2

2 2 2

2 2

cos sin 1cos sin 1 so that

sin sin sinthat is:

cot 1 cosec

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Beyond Pythagoras

(added by John Barnden)Ignore the outer triangle.

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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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.

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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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Switiching to Programme F10 briefly –

reminder of part of Bohnet’s coverage in Term 1

Programme F10: Functions

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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:

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Rotation


By continuing to rotate the radius of a unit circle the trigonometric ratios can extended into the trigonometric functions, valid for any angle:

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Rotation


The sine function:

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Rotation


The cosine function:

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The tangent


The tangent is the ratio of the sine to the cosine:sintancos

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Switching back to Programme F8


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Angles




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Sums and differences of angles

Double angles

Sums and differences of ratios

Products of ratios


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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 tantan( )

1 tan tan

tan tantan( )

1 tan tan

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Double angles


2 2

2

2

2

cos2 cos sin

cos2 1 2sin

cos2 2cos 1

sin 2 2sin cos

2 tantan 2

1 tan

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Sums and differences of trig functions


sin sin 2sin cos2 2

sin sin 2cos sin2 2

cos cos 2cos cos2 2

cos cos 2sin sin2 2

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Products of trig functions


2sin cos sin( ) sin( )

2cos cos cos( ) cos( )

2sin sin cos( ) cos( )

Documents

STROUD Worked examples and exercises are in the text PROGRAMME F8 TRIGONOMETRY