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Engineering MATHEMATICS MET 3403 1.Trigonometric Functions Every right-angled triangle contains two acute angles. With respect to each of these angles, there are six functions, called trigonometric functions, each involving the lengths of two of the sides of the triangle. Consider the following triangle ABC B hypotenuse opposite a C adjacent A AC is the side adjacent to angle a, BC is the side opposite to angle a. Similarly, BC is the side adjacent to angle b, AC is the side opposite to angle b. Six trigonometric functions with respect to angle a: B Hypotenuse (r) opposite (y) a C Adjacent (x) opposite BC y hypotenuse AB r adjacent AC x cos(a ) hypotenuse AB r opposite BC y tan( a ) adjacent AC x sin( a ) Note: sin( a) cos(a) tan( a) A hypotenuse AB 1 opposite BC sin( a) hypotenuse AB 1 sec(a) adjacent AC cos(a) adjacent AC 1 cot( a) opposite AB tan( a) csc(a) cos(a) cot(a) sin( a) Example: Consider the right-angled triangle, with lengths of sides indicated, find sin(d), cos(d), tan(d), sin(e), cos(e), tan(e). E e 13 12 d F EF 12 sin( d ) ED 13 DF 5 cos(d ) ED 13 EF 12 tan( d ) DF 5 5 D DF 5 sin( e) ED 13 EF 12 cos(e) ED 13 DF 5 tan( e) EF 12 Pythagorean Theorem (畢氏定理) B C A BC 2 AC 2 AB 2 Pythagorean Identities derived from Pythagorean theorem sin 2 ( a ) cos 2 ( a ) 1 tan 2 (a ) 1 sec 2 (a ) cot 2 ( a ) 1 csc 2 (a ) Example: In right-angled triangle, sin(a)=4/5, find the values of the other five trigonometric functions of a. B 5 4 a A Since C sin(a)=opposite over hypotenuse=4/5 BC 4, AB 5 AC AB2 BC 2 52 42 3 3 4 3 5 5 cos(a) , tan( a) , cot(a) , sec(a) , csc(a) 5 3 4 3 4 Example: If, in right-angled triangle, sin(a)=7/9, find the values of cos(a) and tan(a). Using trigonometric identity, sin 2 (a) cos 2 (a) 1 2 7 cos 2 (a) 1 cos(a) 9 sin( a) Since tan( a) cos(a) 79 7 7 2 tan( a) 8 4 2 9 4 2 32 4 2 81 9 Angle in degree Each degree is divided into 60 minutes Each minute is divided into 60 seconds Example: Express the angle 265.46 in Degree-Minute-Second (DMS) notation 60' 1 60" 265 27'0.6' 1' 26527'36" 265.46 265 0.46 Angle in radian A unit circle has a circumference of 2 One complete rotation measures 2 radian Angle of 360 = 2 radian Example: 30 30 180 6 180 45 4 4 Special Angles (1) For a 30-60-90 right-angled triangle B 60◦ 2 1 30◦ A From 3 C the triangle, sin( 30 ) sin( 60 ) 1 3 1 3 , cos(30 ) , tan( 30 ) 2 2 3 3 3 1 , cos(60 ) , tan( 60 ) 2 2 3 Special Angles (2) For a 45-45-90 right-angled triangle B 45◦ 2 1 45◦ A From 1 C the triangle, sin( 45 ) cos( 45 ) 1 2 , tan( 45 ) 1 2 2 Unit circle and sine, cosine functions Start measuring angle from positive x-axis ‘+’ angle = anticlockwise θ ‘’ angle = clockwise Angle and quadrants of a trigonometric function for an angle in 2nd, 3rd or 4th quadrants is equal to plus or minus of the value of the 1st quadrant reference angle Value y (x,y) ’ 0 (x,y) ’ Quadrant IIQuadrant I x Quadrant IIIQuadrant IV y y (x,y) ’0 (x,y) ’ (x,y) x ’ 0 ’ (x,y) x The sign of the value is dependent upon the quadrant that the angle is in. Quadrant II SINE +ve Quadrant I ALL +ve Quadrant III TANGENT +ve Quadrant IV COSINE +ve Exercise: find WITHOUT calculator: sin(30 °) cos(45 °) tan(315 °) sin(60 °) cos(180 °) tan(135 °) sin(240 °) cos(-45 °) = _________ = _________ = _________ = _________ = _________ = _________ = _________ = _________ Hint 1 3 1 3 sin( 30 ) , cos(30 ) , tan( 30 ) 2 2 3 3 sin( 60 ) 3 1 , cos(60 ) , tan( 60 ) 3 2 2 •Simple trigonometric equations Notation : If sin = k then = sin-1k (sin-1 is written as inv sin or arcsin). Similar scheme is applied to cos and tan. e.g. Without using a calculator, solve sin = 0.5, where 0o 360o e.g. Solve cos 2 = 0.4 ,where 0 2