‘Trigonometry…the way to do it’ The Author R.M. O’Toole B.A., M.C., M.S.A., C.I.E.A. The author is an experienced state examiner, teacher, lecturer, consultant and author in mathematics. She is an Affiliate Member of the Chartered Institute of Educational Assessors and a Member of The Society of Authors. First Published in 1996 (Revised in 2012) by: Mathematics Publishing Company, ‘The Cottage’ 45, Blackstaff Road, Clough, Downpatrick, Co. Down, BT30 8SR. Tel. 028 44 851818 Email:
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CONTENTS: PART 1:
GCSE COLOURED BLUE;
GCE COLOURED PLUM
RIGHT – ANGLED TRIANGLES
Sin(e), Cos(ine) and Tan(gent) Complementary Angles Trigonometric Ratios for 30, 60 and 45 Given one trigonometric ratio, how to find the others Sample Questions – Right-angled Triangles Angles of Elevation and Depression Bearings - Sample Questions – Bearings Three-dimensional Problems – Angle between a line and a plane – Angle between two planes – Worked Example on 3-dimensional Trigonometry Exercise 1 – Right-angled Triangles Exercise 1 – Right-angled Triangles – Worked Answers
PART 2:
ANGLES OF ANY SIZE
ANGLES BETWEEN 0 AND 360; SINE RULE; COSINE RULE Sin x, Cos x & Tan x (CIRCULAR FUNCTIONS) – RADIANS - GRAPHS Angles between 0° and 360° - ‘CAST’ diagram Sine Rule and Cosine Rule - to decide which rule is applicable Circular Functions (Sin x, Cos x and Tan x) Radians Graphs of Circular Functions (Sin x, Cos x and Tan x) Exercise 2 – Angles of Any Size Exercise 2 – Angles of Any Size – Worked Answers
PART 3
SEC(ant), COSEC(ant) & COT(angent) – Graphs - properties INVERSE TRIGONOMETRIC FUNCTIONS (Arcsin, Arccos & Arctan) - Sin 1 x, Cos 1 x, Tan 1 x – Graphs and Properties
Sec(ant), Cosec(ant) and Cot(angent) - graphs and properties Inverse Trigonometric Functions: Sin 1 x, Cos 1 x and Tan 1 x (Arcsin, Arccos and Arctan) - graphs and properties
PART 4
PYTHAGOREAN IDENTITIES SOLUTION OF SIMPLE TRIGONOMETRIC EQUATIONS .
Pythagorean Identities
sin x cos x
Tan x
Sin 2 x + Cos 2 x 1 - proofs
Sec 2 x 1 + Tan 2 x Cosec 2 x 1 + Cot 2 x - proofs
- Summary of Pythagorean Identities - Worked Examples using Pythagorean Identities - Exercise: Pythagorean Identities - Exercise: Pythagorean Identities – Worked Answers Area of Triangle:
A=
1 ab Sin C 2
- Worked Examples, using formula A =
1 ab Sin C 2
Solution of simple trigonometric equations in a given integral - Worked Examples Trigonometric ratios for 30°, 60° and 45° Polar Co-ordinates: (r, θ) Exercise (with full worked answers) – Polar Co-ordinates Compound Angle Formulae - Sin (A ± B), Cos (A ± B) and Tan (A ± B) Double Angle Formulae - Sin 2A, Cos 2A and Tan 2A (r, α) Formulae Exercise (with full worked answers)
PART 5
CALCULUS:
DIFFERENTIATION; INTEGRATION
TRIGONOMETRIC FUNCTIONS: GRAPHS AND PROPERTIES DIFFERENTIATION INTEGRATION
. Sin(e), Cos(ine) & tan(gent) – Properties and Graphs Sec(ant), Cosec(ant) & Cot(angent) – Properties and Graphs Inverse Trigonometric Functions (Arcsin, Arccos & Arctan) - Sin 1 x, Cos 1 x, Tan 1 x - Properties and Graphs
Differentiation of Trigonometric Functions:
y = sin x;
dy dx
= cosx.
y = cos x;
dy dx
= - sin x.
y = tan x;
dy dx
= sec 2 x.
y = cosec x;
dy dx
= - cosec x cot x.
y = sec x;
dy dx
= sec x tan x.
y = cot x;
dy dx
= - cosec 2 x.
y = sin
y = cos
y = tan
1
1
1
x (arcsin x) x (arccos x) x (arctan x)
1 dy = ; dx 1 x2 1 dy = ; dx 1 x2 1 dy = ; 2 dx x 1
Practice Questions:
Differentiation
Practice Questions:
Differentiation:
x<1 x<1 xєR
Answers
Integration of Trigonometric Functions:
y = sin x:
y dx = - cosx + C.
y = cosx :
y dx = sinx + C.
y = tan x:
y dx = ln
sec x
+ C or – ln
Practice Questions:
Integration
Practice Questions:
Integration: Answers
N.B.
cos x
+ C.
GCE ADVANCED LEVEL SECTIONS 06, 07 AND 08 CONTAIN OUR FULL STUDY OF THE CALCULUS. YOU ARE ADVISED TO USE THEM IN CONJUNCTION WITH TRIGONOMETRY PART 5.
Sample 1:
. . . The general notation for a trigonometric right-angled triangle is as follows:
opposite
hypotenuse xď&#x201A;° adjacent
Diagram 1
. . . Sample 2:
. . . Eg. If the angle of elevation from a point P on the ground to the top of a tower, T, whose base is 100m from P, is 30ď&#x201A;°, the diagram looks like this: T
Tower
Angle of Elevation 30ď&#x201A;° P 100m
With the aid of trigonometry, it is possible to find the height of the tower and the distance PT. . . .
Sample 3:
. . . Solve the following equations: (i)
sin(x - )
=
2
3 4
for 0 x 2.
. . . Method: 3 4
for 0 x 2
sin(x - )
=
sin 1 ( 3 )
=
x-
0.848 +
or
( - 0.848) +
(i)
2
4
2
2
=
2
x
. . . Sample 4:
1.
Cartesian Coordinates (x, y) of a point P (4, 3): y
P (4, 3)
θ
y
(0, 0)
2.
Polar Coordinates (r, θ) of the same point P: P (x = r cos θ, y = r sin θ)
r=5 θ
O
0°