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  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. B CA hypotenuse a adjacent opposite

Six trigonometric functions with respect to angle a: Note: B CA Hypotenuse (r) a Adjacent (x) opposite (y)

Example: Consider the right-angled triangle, with lengths of sides indicated, find sin(d), cos(d), tan(d), sin(e), cos(e), tan(e). FD 13 d 5 12 e E

Pythagorean Theorem ( 畢氏定理 ) B C A Pythagorean Identities  derived from Pythagorean theorem

Example:  In right-angled triangle, sin(a)=4/5, find the values of the other five trigonometric functions of a.  Since sin(a)=opposite over hypotenuse=4/5 A B C 5 4 a

Example:  If, in right-angled triangle, sin(a)=7/9, find the values of cos(a) and tan(a).  Using trigonometric identity,  Since

Angle in degree  Each degree is divided into 60 minutes  Each minute is divided into 60 seconds  Example: Express the angle  in Degree-Minute-Second (DMS) notation Angle in radian  A unit circle has a circumference of 2   One complete rotation measures 2  radian Angle of 360  = 2  radian  Example :

Special Angles (1)  For a right-angled triangle  From the triangle, A B C ◦ 60 ◦

Special Angles (2)  For a right-angled triangle  From the triangle, A B C ◦

Unit circle and sine, cosine functions  Start measuring angle from positive x-axis ‘+’ angle = anticlockwise ‘  ’ angle = clockwise θ

( x,y ) (  x,y )  ’’ ’’ x y 0 ( x,y ) (  x,  y )  ’’ ’’ x y 0 ( x,y ) ( x,  y )  ’’ ’’ x y 0 Quadrant IIQuadrant I Quadrant IVQuadrant III Angle and quadrants  Value of a trigonometric function for an angle in 2 nd, 3 rd or 4 th quadrants is equal to plus or minus of the value of the 1 st quadrant reference angle

Quadrant IIQuadrant I Quadrant IVQuadrant III  The sign of the value is dependent upon the quadrant that the angle is in. ALL +ve SINE +ve COSINE +veTANGENT +ve

Exercise: find WITHOUT calculator: sin(30 °) = _________ cos(45 °)= _________ tan(315 °)= _________ sin(60 °) = _________ cos(180 °)= _________ tan(135 °)= _________ sin(240 °)= _________ cos(-45 °)= _________ Hint

Simple trigonometric equations Notation : If sin  = k then  = sin -1 k (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 0 o    360 o e.g. Solve cos 2  =  0.4,where 0    2 