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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 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 http://fatmuscle.cjb.net 1 Maths Extension 1 – Trigonometry Trigonometric Ratios θ hypotenuse adjacent opposite hypotenuse θ adjacent opposite Sine sin = Cosine cos = Tangent tan = Cosecant cosec = Secant sec = Cotangent cot = sin cos tan cosec sec cot = = = = = = sin cos = = = cos90 sin 90 cot 90 sec90 cosec90 tan 90 60’’ = 1’ 60’ = 1° 60 seconds = 1 minute 60 minutes = 1 degree tan 1 sin 1 cos 1 tan opposite hypotenuse adjacent hypotenuse opposite adjacent hypotenuse opposite hypotenuse adjacent adjacent opposite cot cos sin http://fatmuscle.cjb.net 2 Maths Extension 1 – Trigonometry Exact Values & Triangles 30° 2 1 2 3 45° 60° 1 sin cos tan cos ec sec cot 0° 0 1 0 –– 1 –– 1 30° 60° 45° 1 2 3 2 1 2 1 2 1 2 3 2 1 3 1 3 2 2 3 2 2 3 2 2 1 3 1 sin 2 cos 2 =1 = 1 sin 2 = 1 cos 2 3 90° 1 0 –– 1 –– 0 180° 0 –1 0 –– –1 –– Trigonometric Identities cos 2 sin 2 = cosec2 2 cot 2 = cosec – 1 1 = cosec2 – cot 2 1 cot 2 tan 2 1 = 2 tan = 1 = sec 2 sec 2 1 sec 2 tan 2 http://fatmuscle.cjb.net 3 Maths Extension 1 – Trigonometry ASTC Rule 90° nd 2 Quadrant 1st Quadrant S A T C 180° 3rd Quadrant 0 360 4th Quadrant 270° First Quadrant: All positive sin sin cos cos tan tan + + + Second Quadrant: Sine positive sin 180 sin cos180 – cos tan 180 – tan + – – Third Quadrant: Tangent positive sin 180 – sin cos180 – cos tan 180 tan – – + Fourth Quadrant: Cosine positive sin 360 – sin cos360 cos tan 360 – tan – + – http://fatmuscle.cjb.net 4 Maths Extension 1 – Trigonometry Trigonometric Graphs Sine & Cosine Rules Sine Rule: a b c sin A sin B sin C sin A sin B sin C a b c OR A c b C B a Cosine Rule: a 2 b 2 c 2 2bc cos A A c b http://fatmuscle.cjb.net 5 a Maths Extension 1 – Trigonometry Area of a Triangle A 12 ab sin C C is the angle a & b are the two adjacent sides C b a http://fatmuscle.cjb.net 6 Maths Extension 1 – Trigonometry 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 Solve sin θ = for 0 360 sin = 12 = 30°, 150° 1 2 Example 2 Solve cos 2θ = for 0 360 cos 2 = 12 2 = 60°, 300°, 420°, 660° = 30°, 150°, 210°, 330° 1 2 Example 3 Solve tan 2 = 1 for 0 360 tan 2 = 1 = 45°, 225° 2 = 90° Example 4 sin 2 cos 0 2sin cos cos cos 2 sin 1 =0 =0 cos = 12 = 210°, 330° =0 = 90°, 270° sin Example 5 3sin cos 2 2 3sin 1 2 sin 2 = –2 http://fatmuscle.cjb.net 7 Maths Extension 1 – Trigonometry 2 sin 2 3 sin 1 2sin 1sin 1 sin =0 =0 = 12 = 210°, 330° = –1 = 270° sin http://fatmuscle.cjb.net 8 Maths Extension 1 – Trigonometry 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 tan 1 tan tan tan tan 1 tan tan Double Angles = 2sin cos sin 2 cos 2 = cos 2 sin 2 = 1 2 sin 2 = 2 cos2 1 tan 2 = sin 2 = 12 1 cos 2 = 12 1 cos 2 cos 2 2 tan 1 2 tan 2 Triple Angles = 3sin 4 sin 3 sin 3 = 4 cos3 3 cos cos 3 tan 3 = Half Angles = sin cos 3 tan tan 3 1 3 tan 2 2 sin 2 cos 2 = cos2 2 sin 2 2 = 1 2 sin 2 2 http://fatmuscle.cjb.net 9 Maths Extension 1 – Trigonometry tan = 2 cos 2 2 1 = 2 tan 2 1 2 tan 2 2 http://fatmuscle.cjb.net 10 Maths Extension 1 – Trigonometry Deriving the Triple Angles = sin 2 sin 3 = sin 2 cos cos 2 sin = 2 sin cos cos 1 2 sin 2 sin = 2 sin cos 2 sin 2 sin 3 = 2 sin 1 sin 2 sin 2 sin 3 = 2 sin 2 sin 3 sin 2 sin 3 = 3sin 4 sin 3 cos 3 tan 3 = = = = = = = cos2 = tan 2 tan 2 tan 1 tan 2 tan 2 tan tan 1 tan 2 = = = = _ Normal double angle_ Expand double angle_ Multiply_ Change sin 2 cos 2 1 _ Simplify_ cos 2 cos sin 2 sin 2 cos 1cos 2 sin cos sin 2 2 cos3 cos 2 sin 2 cos 2 cos3 cos 2 1 cos 2 cos 2 cos 2 cos 2 cos 2 cos3 4 cos3 3 cos tan 1 2 1tan tan 2 2 tan tan tan 3 1 tan 2 1 tan 2 2 tan 2 1 tan 2 3 3 tan tan 1 3 tan 2 http://fatmuscle.cjb.net 11 Maths Extension 1 – Trigonometry T – Formulae Let t = tan 2 sin = sin = cos = tan = 2 sin 2 cos 2 = 2t 1 t2 1 t2 1 t2 2t 1 t2 Using half angles _ Divide by “1” 2 sin 2 cos 2 cos 2 2 sin 2 2 sin 2 cos 2 1 = 2 sin 2 cos 2 cos 2 2 cos 2 2 sin 2 2 cos 2 2 Divide top and bottom by cos 2 = = 2 tan 2 1 tan 2 2 2t 1 t2 cos ’ cos = cos 2 2 sin 2 2 = cos 2 2 sin 2 2 cos 2 2 sin 2 2 = sin cos 2 2 cos 2 2 sin 2 2 cos 2 2 = 1 tan 2 2 1 tan 2 2 = 1 t2 1 t2 cos 2 2 tan cancel; = sin cos = 2t 1 t 2 1 t 2 1 t 2 = 2t 1 t2 2 2 sin cos becomes tan 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 Rcos x = = R 2 cos 2 x R 2 sin 2 x b2 a 2 b2 2 2 = sin x cos x 1 R2 Rsin x tan R a 2 b2 a sin x b cos x =C =C =C =C a sin x b cos x a cos x b sin x a cos x b sin x b a R sin( x ) R sin( x ) R cos( x ) R cos( x ) Example 1 Find x. 3 sin x cos x 1 R = 2 tan 3 12 = 4 =2 2 sin( x 30) sin( x 30) x 30 x = 1 3 = 30° =1 = 12 = 30°, 150° = 60°, 180° http://fatmuscle.cjb.net 13 Maths Extension 1 – Trigonometry General Solutions of Trigonometric Equations Then n (1)n sin sin cos cos Then 2n tan tan Then n Radians c = 180° 1° = c 180 Arcs, Sectors, Segments Arc Length l = r l θ r θ r Area of Sector A = 12 r 2 http://fatmuscle.cjb.net 14 Maths Extension 1 – Trigonometry Area of Segment A = 12 r 2 sin Segment θ r http://fatmuscle.cjb.net 15 Maths Extension 1 – Trigonometry Trigonometric Limits sin x tan x lim = lim x 0 x 0 x x = lim cos x x0 =1 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 = a cos( ax b) d cos x dx = sin x d cos f ( x) dx = f ' ( x) sin f ( x) d cos(ax b) dx = a sin( ax b) d tan x dx = sec 2 x d tan f ( x) dx = f ' ( x) sec 2 f ( x) 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 = sec = 1 tan ax c a = x sin 1 c a = x x cos 1 c __OR__ sin 1 c a a = 1 x tan 1 c a a cos ec ax dx = 1 cot ax c a sec ax. tan ax dx = 1 sec ax c a cos ecax.cot ax dx = 1 cos ecax c a ax dx 1 a x 2 dx 2 1 a 2 a x 2 2 2 1 x2 dx dx 2 1 cos ax c a http://fatmuscle.cjb.net 18 Maths Extension 1 – Trigonometry Integration of sin2x and cos2x = 2 cos 2 x 1 cos 2x = 2 cos 2 x cos 2x 1 1 cos 2 x 1 = cos 2 x 2 2 = 12 cos 2 x 1 dx cos x dx = 12 12 sin 2 x x C = 14 sin 2 x 12 x C cos cos 2x 2 sin 2 x sin 2 x sin 2 x dx 2 x dx = 1 4 sin 2 x 12 x C x dx = 1 2 x 14 sin 2 x C = 1 sin 2 x = 1 cos 2x = 12 1 cos 2 x = 12 1 cos 2 x dx = 12 x 12 sin 2 x C = 12 x 14 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 2 x -2 2 2 y 2 2 sin 1 ( x) sin 1 x Inverse Cos – Graph, Domain, Range, Properties y 1 x 1 0 y 2 x -1 0 1 cos 1 ( x) cos 1 x Inverse Tan – Graph, Domain, Range, Properties y 2 All real x 2 x 2 -2 http://fatmuscle.cjb.net 20 Maths Extension 1 – Trigonometry 2 y 2 tan 1 ( x) tan 1 x http://fatmuscle.cjb.net 21 Maths Extension 1 – Trigonometry Differentiation of Inverse Trigonometric Functions d sin 1 x = 1 2 1 x dx d sin 1 ax dx d sin 1 f ( x) dx 1 = a x2 2 f ' ( x) = 1 [ f ( x)]2 = = = d cos 1 x dx d cos 1 ax dx d cos 1 f ( x) dx 1 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 tan 1 ax dx d tan 1 f ( x) dx 2 http://fatmuscle.cjb.net 22 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 1 c a a http://fatmuscle.cjb.net 23