Trigonometry and Inverse Trigonometry

Maths Extension – Trigonometry

Trigonometry  Trigonometric Ratios  Exact Values & Triangles  Trigonometric Identities  All Silver Tea Cups rule  Trigonometric Graphs  Sine & Cosine Rules  Area of a Triangle  Trigonometric Equations  Sums and Differences of angles  Double Angles  Triple Angles  Half Angles  T – formula  Subsidiary Angle formula  General Solutions of Trigonometric Equations  Radians  Arcs, Sectors, Segments  Trigonometric Limits  Differentiation of Trigonometric Functions  Integration of Trigonometric Functions  Integration of sin2x and cos2x  INVERSE TRIGNOMETRY  Inverse Sin – Graph, Domain, Range, Properties  Inverse Cos – Graph, Domain, Range, Properties  Inverse Tan – Graph, Domain, Range, Properties  Differentiation of Inverse Trigonometric Functions  Integration of Inverse Trigonometric Functions

Addu High School/Department of Mathematics/Resources/Math-ebook_Trigonometry Page 1 of 23 Maths Extension – Trigonometry

Trigonometric Ratios

θ hypotenuse hypotenuse e t t i n s e o c p a j p d o a θ adjacent opposite

opposite  Sine sin = hypotenuse adjacent  Cosine cos = hypotenuse opposite  Tangent tan = adjacent 1 hypotenuse Cosecant cosec = = sin opposite 1 hypotenuse Secant sec = = cos adjacent 1 adjacent Cotangent cot = = tan opposite sin = cos90   cos = sin90   tan = cot90   cosec = sec90   sec = cosec90   cot = tan90  

60 seconds = 1 minute 60’’ = 1’ 60 minutes = 1 degree 60’ = 1°

sin cos tan  cot  cos sin

Exact Values & Triangles

30° 2 1 3 2

45° 60° 1 1

0° 30° 60° 45° 90° 180° 1 3 1 sin 0 2 2 2 1 0 cos 3 1 1 1 2 2 2 0 –1 1 tan 0 3 3 1 –– 0 cos 2 ec –– 2 3 2 1 –– sec 2 1 3 2 2 –– –1 1 cot –– 3 3 1 0 ––

Trigonometric Identities sin 2   cos2  = 1 cos2  = 1 sin 2  sin 2  = 1 cos2 

1 cot2  = cosec2 cot 2  = cosec2 – 1 1 = cosec2 – cot 2 

tan2  1 = sec2  tan2  = sec2  1 1 = sec2   tan2 

All Silver Tea Cups Rule

90° nd 2 Quadrant 1st Quadrant S A 180° 0 360 3rd Quadrant T C 4th Quadrant

First Quadrant: All positive sin sin + cos cos + tan tan +

Second Quadrant: Sine positive sin180   sin + cos180   –cos – tan180   – tan –

Third Quadrant: Tangent positive sin180   –sin – cos180   –cos – tan180   tan +

Fourth Quadrant: Cosine positive sin360   –sin – cos360   cos + tan360   – tan –

Trigonometric Graphs

Sine & Cosine Rules Sine Rule: a b c sin A sin B sinC   OR   sin A sin B sinC a b c

Cosine Rule: a 2  b 2  c 2  2bc cos A

a Area of a Triangle 1 A  2 absinC  C is the angle  a & b are the two adjacent sides

Trigonometric Equations  Check the domain eg. 0    360  Check degrees (0    360) or radians (0    2 )  If double angle, go 2 revolutions  If triple angle, go 3 revolutions, etc…  If half angles, go half or one revolution (safe side)

Example 1 1 Solve sin θ = 2 for 0    360 sin 1 = 2  = 30°, 150°

Example 2 1 Solve cos 2θ = 2 for 0    360 cos2 1 = 2 2 = 60°, 300°, 420°, 660°  = 30°, 150°, 210°, 330°

Example 3  Solve tan 2 = 1 for 0    360  tan 2 = 1  2 = 45°, 225°  = 90°

Example 4 sin 2  cos  0 2sin cos  cos = 0 cos 2sin 1 = 0

cos 1 = 0 sin =  2  = 90°,  = 210°, 270° 330°

Example 5 3sin  cos2  2 3sin  1 2sin 2   = –2 2sin 2   3sin 1 = 0

2sin 1sin 1 = 0

sin 1 =  2 sin = –1  = 210°,  = 270° 330°

Sums and Differences of angles sin    = sin cos   cos sin  sin    = sin cos   cos sin  cos    = cos cos   sin sin  cos    = cos cos   sin sin  tan  tan  tan    = 1 tan tan  tan  tan  tan    = 1 tan tan 

Double Angles sin 2 = 2sin cos cos2 = cos2   sin 2  = 1 2sin 2  = 2cos2  1 tan 2 2tan = 2 1 2tan 

2 1 sin  = 2 1 cos2  2 1 cos  = 2 1 cos2 

Triple Angles sin3 = 3sin  4sin3  cos3 = 4cos3   3cos 3tan  tan3  tan3 = 1 3tan2 

Half Angles sin   = 2sin 2 cos 2 cos 2  2  = cos 2  sin 2 2  = 1 2sin 2 2  = 2cos 2 1

Addu High School/Department of Mathematics/Resources/Math-ebook_Trigonometry Page 9 of 23 Maths Extension – Trigonometry tan  2tan 2 = 2  1 2tan 2

Deriving the Triple Angles sin3 = sin2   _ = sin 2 cos  cos2 sin Normal double angle_ = 2sin cos cos  1 2sin2  sin Expand double angle_ = 2sin cos2   sin  2sin3  Multiply_ = 2sin 1 sin 2   sin  2sin3 Change sin 2   cos2   1 = 2sin  2sin3  sin  2sin3  _ = 3sin  4sin3  Simplify_ cos3 = cos2   = cos2 cos  sin 2 sin = 2cos2  1cos  2sin cos sin = 2cos3   cos  2sin 2  cos = 2cos3   cos  21 cos2  cos = 2cos2   cos  2cos  2cos3  = 4cos3   3cos tan3 = tan2   tan 2  tan = 1 tan 2 tan 2 tan   tan 1tan 2  = 1 2 tan  tan  1tan 2  2 tan  tan  tan 3  1tan 2  = 1tan 2  2 tan 2  1tan 2  3 = 3tan  tan  1 3tan2 

T – Formulae  Let t = tan 2 2t sin = 2 1 t 1 t 2 cos = 1 t 2 2t tan = 2 1 t sin   = 2sin 2 cos 2 Using half angles   2sin 2 cos 2 _ = 2  2  cos 2  sin 2 Divide by “1” sin 2   cos2   1   2sin 2 cos 2 2  cos 2 = 2  2  cos 2  sin 2 2  cos 2 Divide top and bottom by cos2   2tan 2 = 2  1 tan 2 2t = 2 1 t cos sin ’ cancel; cos becomes tan cos cos2   sin 2  tan sin = 2 2 = 2  2  cos cos 2  sin 2 = cos2   sin2  2 2 2t 1t 2 = 1t 2 2 2   1t 2 cos 2  sin 2 2  cos 2 = cos2   sin 2  2t 2 2 = 2 2  1 t cos 2

2  1 tan 2 = 2  1 tan 2 2 = 1 t 1 t 2

Subsidiary Angle Formula asin x  bcos x = R(sin xcos x  cos xsin x) = Rsin xcos x  Rcos xsin x

a 2 2 2 = Rcos x  a = R cos x b = Rsin x b2 = R2 sin 2 x a2  b2 sin 2 x  cos2 x  1 = R2

b R  a2  b2 tan  a asin x  bcos x = C Rsin(x ) asin x  bcos x = C Rsin(x ) acos x  bsin x = C Rcos(x ) acos x  bsin x = C Rcos(x )

Example 1 Find x. 3sin x  cos x  1

1 R = 2 2 tan = 3 1 3 = 4  = 30° = 2 2sin(x  30) = 1 sin(x  30) 1 = 2 x  30 x = 30°, 150° = 60°, 180°

General Solutions of Trigonometric Equations sin  sin Then   n  (1)n cos  cos Then   2n  tan  tan Then   n 

Radians  c = 180° c 1° =  180

Arcs, Sectors, Segments Arc Length l = r l

Area of Sector 1 2 A = 2 r 

Area of Segment 1 2 A = 2 r   sin  Segment

Trigonometric Limits sin x tan x lim = lim = limcos x = 1 x0 x x0 x x0

Differentiation of Trigonometric Functions d sin x dx = cos x d sin f (x) dx = f '(x)cos f (x) d sin(ax  b) dx = acos(ax  b) d cos x dx =  sin x d cos f (x) dx =  f '(x)sin f (x) d cos(ax  b) dx =  asin(ax  b) d tan x dx = sec2 x d tan f (x) dx = f '(x)sec2 f (x) d tan(ax  b) dx = asec2 (ax  b) d sec x dx = sec x.tan x d cosecx dx =  cot x.cosecx d 2 cot x =  cosec x dx

Integration of Trigonometric Functions

1 cosax dx = sin ax  c  a

1 sin ax dx =  cosax  c  a

1 sec2 ax dx = tan ax  c  a

1  x  sin 1  c  2 2 dx =   a  x  a 

1  x   x   cos1  c  sin 1  c  2 2 dx =   __OR__   a  x  a   a 

1 1  x  tan1  c 2 2 dx =    a  x a  a 

1 cosec2ax dx =  cot ax  c  a

1 secax.tan ax dx = secax  c  a

1 cosecax.cot ax dx =  cosecax  c  a

Integration of sin 2 x and cos 2x cos2x = 2cos2 x 1 cos2x 1 = 2cos2 x 1 cos2x 1 2 = cos2 x cos2 x 1 cos2x 1  dx = 2   dx 1 1 = 2 2 sin 2x  x C 1 1 = 4 sin 2x  2 x  C

2 1 sin 2x  1 x  C cos x dx = 4 2 cos2x = 1 sin 2 x 2 2sin x 1 cos2x 2 = sin x 1 = 2 1 cos2x sin 2 x 1 1 cos2x  dx = 2   dx 1 1 = 2 x  2 sin 2x C 1 1 = 2 x  4 sin 2x  C

2 1 x  1 sin 2x  C sin x dx = 2 4

INVERSE TRIGNOMETRY Inverse Sin – Graph, Domain, Range, Properties

y 1  x  1  2

x     y  -2 2 2 2

  2

sin 1(x)  sin 1 x

Inverse Cos – Graph, Domain, Range, Properties y 1  x  1 

 0  y   2

x 0 -1 1

cos1(x)    cos1 x

Inverse Tan – Graph, Domain, Range, Properties y 2  All real x 2

x     y  2 2

  2 -2

tan1(x)   tan1 x

Differentiation of Inverse Trigonometric Functions d 1 sin1 x = dx 1 x2 d 1 sin1 x  a  = dx a2  x2 d f '(x) sin1 f (x) dx = 1[ f (x)]2 d 1 cos1 x =  dx 1 x2 d 1 cos1 x   a  = dx a2  x2 d f '(x) cos1 f (x)  dx = 1[ f (x)]2

d 1 1 tan x = 2 dx 1 x

d 1 x a tan a  = 2 2 dx a  x d f '(x) tan1 f (x) dx = a  [ f (x)]2

Integration of Inverse Trigonometric Functions

1  x  sin 1  c  2 2 dx =   a  x  a 

1  x   x   cos1  c  sin 1  c  2 2 dx =   __OR__   a  x  a   a 

1 1  x  tan1  c 2 2 dx =    a  x a  a 

Addu High School/Department of Mathematics/Resources/Math-ebook_Trigonometry Page 23 of 23