trigonometry

Maths Extension 1 – Trigonometry Trigonometry  Trigonometric Ratios  Exact Values & Triangles  Trigonometric Identities  ASTC 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 sin 2 x and cos 2 x  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 http://fatmuscle.cjb.net 1 Maths Extension 1 – Trigonometry Trigonometric Ratios Sine sin θ = hypotenuse opposite Cosine cos θ = hypotenuse adjacent Tangent tan θ = adjacent opposite Cosecant cosec θ = θ sin 1 = opposite hypotenuse Secant sec θ = θ cos 1 = adjacent hypotenuse Cotangent cot θ = θ tan 1 = opposite adjacent sin θ = ( ) θ −°90cos cos θ = ( ) θ −°90sin tan θ = ( ) θ −°90cot cosec θ = ( ) θ −°90sec sec θ = ( ) θ −°90cosec cot θ = ( ) θ −°90tan 60 seconds = 1 minute 60’’ = 1’ 60 minutes = 1 degree 60’ = 1° θ θ θ cos sin tan = θ θ θ sin cos cot = http://fatmuscle.cjb.net 2 θ θ hypotenuse hypotenuse opposite adjacent adjacent opposite Maths Extension 1 – Trigonometry Exact Values & Triangles 0° 30° 60° 45° 90° 180° sin 0 2 1 2 3 2 1 1 0 cos 1 2 3 2 1 2 1 0 –1 tan 0 3 1 3 1 –– 0 cos ec –– 2 3 2 2 1 –– sec 1 3 2 2 2 –– –1 cot –– 3 3 1 1 0 –– Trigonometric Identities θθ 22 cossin + = 1 θ 2 cos = θ 2 sin1− θ 2 sin = θ 2 cos1− θ 2 cot1+ = cosec 2 θ θ 2 cot = cosec 2 θ – 1 1 = cosec 2 θ – θ 2 cot 1tan 2 + θ = θ 2 sec θ 2 tan = 1sec 2 − θ 1 = θθ 22 tansec − http://fatmuscle.cjb.net 3 1 1 2 45° 3 1 2 30° 60° Maths Extension 1 – Trigonometry ASTC Rule First Quadrant: All positive θ sin θ sin + θ cos θ cos + θ tan θ tan + Second Quadrant: Sine positive ( ) θ −°180sin θ sin + ( ) θ −°180cos – θ cos – ( ) θ −°180tan – θ tan – Third Quadrant: Tangent positive ( ) θ +°180sin – θ sin – ( ) θ +°180cos – θ cos – ( ) θ +°180tan θ tan + Fourth Quadrant: Cosine positive ( ) θ −°360sin – θ sin – ( ) θ −°360cos θ cos + ( ) θ −°360tan – θ tan – http://fatmuscle.cjb.net 4 ° ° 360 0 90° 180° 270° S A T C 1 st Quadrant 4 th Quadrant 2 nd Quadrant 3 rd Quadrant Maths Extension 1 – Trigonometry Trigonometric Graphs Sine & Cosine Rules Sine Rule: C c B b A a sinsinsin == OR c C b B a A sinsinsin == Cosine Rule: Abccba cos2 222 −+= http://fatmuscle.cjb.net 5 A B C a b c A a b c Maths Extension 1 – Trigonometry Area of a Triangle CabA sin 2 1 =  C is the angle  a & b are the two adjacent sides http://fatmuscle.cjb.net 6 C b a Maths Extension 1 – Trigonometry Trigonometric Equations  Check the domain eg. °≤≤° 3600 θ  Check degrees ( °≤≤° 3600 θ ) or radians ( πθ 20 ≤≤ )  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 Solve sin θ = 2 1 for °≤≤° 3600 θ θ sin = 2 1 θ = 30°, 150° Example 2 Solve cos 2θ = 2 1 for °≤≤° 3600 θ θ 2cos = 2 1 θ 2 = 60°, 300°, 420°, 660° θ = 30°, 150°, 210°, 330° Example 3 Solve tan 2 θ = 1 for °≤≤° 3600 θ tan 2 θ = 1 2 θ = 45°, 225° θ = 90° Example 4 0cos2sin =+ θθ θθθ coscossin2 + = 0 ( ) 1sin2cos + θθ = 0 θ cos = 0 θ sin = 2 1 − θ = 90°, 270° θ = 210°, 330° Example 5 22cossin3 −=− θθ ( ) θθ 2 sin21sin3 −− = –2 1sin3sin2 2 ++ θθ = 0 http://fatmuscle.cjb.net 7 Maths Extension 1 – Trigonometry ( )( ) 1sin1sin2 ++ θθ = 0 θ sin = 2 1 − θ sin = –1 θ = 210°, 330° θ = 270° http://fatmuscle.cjb.net 8 Maths Extension 1 – Trigonometry Sums and Differences of angles ( ) βα +sin = βαβα sincoscossin + ( ) βα −sin = βαβα sincoscossin − ( ) βα +cos = βαβα sinsincoscos − ( ) βα −cos = βαβα sinsincoscos + ( ) βα +tan = βα βα tantan1 tantan − + ( ) βα −tan = βα βα tantan1 tantan + − Double Angles θ 2sin = θθ cossin2 θ 2cos = θθ 22 sincos − = θ 2 sin21− = 1cos2 2 − θ θ 2tan = θ θ 2 tan21 tan2 − θ 2 sin = ( ) θ 2cos1 2 1 − θ 2 cos = ( ) θ 2cos1 2 1 + Triple Angles θ 3sin = θθ 3 sin4sin3 − θ 3cos = θθ cos3cos4 3 − θ 3tan = θ θθ 2 3 tan31 tantan3 − − Half Angles θ sin = 22 cossin2 θθ θ cos = 2 2 2 2 sincos θθ − = 2 2 sin21 θ − = 1cos2 2 2 − θ http://fatmuscle.cjb.net 9 Maths Extension 1 – Trigonometry θ tan = 2 2 2 tan21 tan2 θ θ − http://fatmuscle.cjb.net 10 [...]... http://fatmuscle.cjb.net 13 Maths Extension 1 – Trigonometry General Solutions of Trigonometric Equations sin θ = sin α Then θ = nπ + (−1) nα cosθ = cosα Then θ = 2nπ ± α tan θ = tan α Then θ = nπ + α Radians πc = 180° 1° = πc 180 Arcs, Sectors, Segments Arc Length l = rθ l θ r Area of Sector A = 1 r 2θ 2 θ r http://fatmuscle.cjb.net 14 Maths Extension 1 – Trigonometry Area of Segment A = 1 r 2 (θ − sin... d ( tan(ax + b) ) dx = a sec 2 (ax + b) d sec x dx = sec x tan x d cos ecx dx = − cot x cos ecx = − cos ec 2 x d cot x dx http://fatmuscle.cjb.net 16 Maths Extension 1 – Trigonometry http://fatmuscle.cjb.net 17 Maths Extension 1 – Trigonometry Integration of Trigonometric Functions ∫ cos ax dx = 1 sin ax + c a ∫ sin ax dx = 1 − cos ax + c a ∫ sec = 1 tan ax + c a =  x sin −1   + c a =  x ... 2 + sin 2 θ 2 2 θ cos 2 = = 2t 1+ t 2 1− t 2 1+ t 2 = becomes tan sin θ cosθ = sin cos 2t 1− t2 1 − tan 2 θ 2 1 + tan 2 θ 2 = tan θ cancel; 1− t2 1+ t2 http://fatmuscle.cjb.net 12 Maths Extension 1 – Trigonometry Subsidiary Angle Formula a sin x + b cos x a b = = = = R (sin x cos x + cos x sin x ) R sin x cos x + R cos x sin x ∴ a2 R cos x = = R 2 cos 2 x ∴ b2 R 2 sin 2 x a 2 + b2 sin 2 x + cos 2 x...Maths Extension 1 – Trigonometry Deriving the Triple Angles sin 3θ = sin ( 2θ + θ ) = sin 2θ cosθ + cos 2θ sin θ = 2 sin θ cosθ cosθ + (1 − 2 sin 2 θ ) sin θ = 2 sin θ cos2 θ + sin θ − 2 sin 3 θ = 2 sin θ (1 − sin 2 θ ) +... l θ r Area of Sector A = 1 r 2θ 2 θ r http://fatmuscle.cjb.net 14 Maths Extension 1 – Trigonometry Area of Segment A = 1 r 2 (θ − sin θ ) 2 Segment θ r http://fatmuscle.cjb.net 15 Maths Extension 1 – Trigonometry Trigonometric Limits lim x →0 sin x x = lim x→0 tan x x = lim cos x x→0 =1 Differentiation of Trigonometric Functions d ( sin x ) dx = cos x d [ sin f ( x)] dx = f ' ( x) cos f ( x) d ( sin(ax... θ + tan θ 1− tan 2 θ tan θ 1 − 2 1− tan tan θ 2 θ 2 tan θ + tan θ − tan 3 θ 1− tan 2 θ 1− tan 2 θ − 2 tan 2 θ 1− tan 2 θ 3 3 tan θ − tan θ 1 − 3 tan 2 θ http://fatmuscle.cjb.net 11 Maths Extension 1 – Trigonometry T – Formulae Let t = tan θ2 sin θ cosθ = = = tan θ sin θ = = 2t 1+ t2 1− t2 1+ t2 2t 1− t2 2 sin θ cos θ 2 2 θ 2 2 sin cos cos 2 θ + sin 2 Using half angles _ Divide by “1” θ 2 2 θ 2 sin 2... 1 − cot ax + c a ∫ sec ax tan ax dx = 1 sec ax + c a ∫ ax dx 1 a −x 2 dx 2 1 ∫− ∫a 2 a −x 2 2 1 + x2 2 dx dx 2 ∫ cos ecax.cot ax dx = 1 − cos ecax + c a http://fatmuscle.cjb.net 18 Maths Extension 1 – Trigonometry Integration of sin2x and cos2x cos 2 x = 2 cos2 x − 1 cos 2 x + 1 = 2 cos2 x 1 ( cos 2 x + 1) = cos2 x 2 2 = 1 ∫ ( cos 2 x + 1) dx 2 ∫ cos x dx 1 1 = 2 ( 2 sin 2 x + x ) + C = 1 sin 2 x +... sin 2 x + C 4 = 1 − sin 2 x = 1 − cos 2 x = 1 (1 − cos 2 x ) 2 1 = 2 ∫ (1 − cos 2 x ) dx = 1 ( x − 1 sin 2 x ) + C 2 2 1 1 = 2 x − 4 sin 2 x + C ∫ sin 2 http://fatmuscle.cjb.net 19 Maths Extension 1 – Trigonometry INVERSE TRIGNOMETRY Inverse Sin – Graph, Domain, Range, Properties y −1 ≤ x ≤ 1 π 2 x -2 − 2 π π ≤y≤ 2 2 −π 2 sin −1 (− x) = − sin −1 x Inverse Cos – Graph, Domain, Range, Properties y −1... x -1 0 1 cos −1 (− x) = π − cos −1 x Inverse Tan – Graph, Domain, Range, Properties y 2 π 2 x − −π 2 -2 All real x π π ≤y≤ 2 2 tan −1 (− x) = − tan −1 x http://fatmuscle.cjb.net 20 Maths Extension 1 – Trigonometry Differentiation of Inverse Trigonometric Functions ( ) = ( ) = d sin −1 x dx d x sin −1 a dx ( d sin −1 f ( x) dx ) 1 1 − x2 1 a − x2 2 f ' ( x) = 1 − [ f ( x)]2 ( ) = − ( ) = − = − d cos... 1 − x2 1 a2 − x2 f ' ( x) 1 − [ f ( x)]2 ( ) = 1 1 + x2 ( ) = a a + x2 = f ' ( x) a + [ f ( x)]2 d tan −1 x dx d x tan −1 a dx ( d tan −1 f ( x) dx ) 2 http://fatmuscle.cjb.net 21 Maths Extension 1 – Trigonometry Integration of Inverse Trigonometric Functions ∫ 1 a −x 2 1 ∫− ∫a dx 2 a2 − x2 2 1 + x2 dx dx =  x sin −1   + c a =  x  x cos −1   + c OR − sin −1   + c a a = 1 x tan