STROUD Worked examples and exercises are in the text Programme F9: Trigonometry PROGRAMME F9...
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Transcript of STROUD Worked examples and exercises are in the text Programme F9: Trigonometry PROGRAMME F9...
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
PROGRAMME F9
TRIGONOMETRY
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Trigonometric identities
Trigonometric formulas
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Trigonometric identities
Trigonometric formulas
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Rotation
Radians
Triangles
Trigonometric ratios
Reciprocal ratios
Pythagoras’ theorem
Special triangles
STROUD
Worked examples and exercises are in the text
Programme F9: 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 – 360°
One degree = 60 minutes and one minute = 60 seconds
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
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
STROUD
Worked examples and exercises are in the text
Programme F9: 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 BCA B A C B C
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Trigonometric ratios
so that:
and and
AB AC BCA B A C B C
AB A B AB A B AC A CAC A C BC B C BC B C
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Trigonometric ratios
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Reciprocal ratios
1 of angle - denoted by cosecsin
1 of angle - denoted by seccos
1 of angle - denoted by cottan
cosecant
secant
cotangent
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Pythagoras’ 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
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Special triangles
Right-angled isosceles
Angles measured in degrees:1sin 45 cos45 and tan 45 12
Angles measured in radians:1sin / 4 cos / 4 and tan / 4 12
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Special triangles
Half equilateral
Angles measured in degrees:1sin30 cos602
3sin 60 cos30 2
1tan 60 3tan30
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Special triangles
Half equilateral
Angles measured in radians:1sin / 6 cos /32
3sin /3 cos / 62
1tan /3 3tan / 6
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Trigonometric identities
Trigonometric formulas
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Trigonometric identities
Trigonometric formulas
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Trigonometric identities
The fundamental identity
Two more identities
Identities for compound angles
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Trigonometric identities
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 cc c
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Trigonometric identities
Two more identities
Dividing the fundamental identity by
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Trigonometric identities
Two more identities
Dividing the fundamental identity by
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Trigonometric identities
Trigonometric formulas
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Angles
Trigonometric identities
Trigonometric formulas
STROUD
Worked examples and exercises are in the text
Programme F9: 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 F9: Trigonometry
Trigonometric formulas
Sums and differences of angles
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Trigonometric formulas
Double angles
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Trigonometric formulas
Sums and differences of ratios
sin sin 2sin cos2 2
sin sin 2cos sin2 2
cos cos 2cos cos2 2
cos cos 2sin sin2 2
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Trigonometric formulas
Products of ratios
2sin cos sin( ) sin( )
2cos cos cos( ) cos( )
2sin sin cos( ) cos( )
STROUD
Worked examples and exercises are in the text
Programme F9: Trigonometry
Learning outcomes
Convert angles measured in degrees, minutes and seconds into decimal degrees
Convert degrees into radians and vice versa
Use a calculator to determine the values of trigonometric ratios for any acute angle
Verify trigonometric identities