Chapter 7 Trigonometry

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Right-angled Triangles Adjacent side The side AB opposite to right angle (i.e. the longest side) is called hypotenuse. Opposite side When  BAC, one of the acute angles, is marked as ,  BC is called the opposite side of , and AC is called the adjacent side of . Hypotenuse

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Hypotenuse sin   Opposite side Trigonometric Ratios (I) Adjacent side Opposite side Hypotenuse  cos   Adjacent side Hypotenuse tan   Adjacent side Opposite side

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Trigonometric Ratios (II)

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Special Angles 30° and 60° B D A 60  2 2 2

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Special Angles 30° and 60° C 1 60  30  B A 2

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Special Angle 45° B C A 45  tan 45   1

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Five Special Angles 30  60  45  90  cos  00 sin  00 30  45  60  90 

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Trigonometric Ratios of Any Angle

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Signs of Trigonometric Ratios sin   0 cos   0 tan   0 Only sin   0 Only tan   0Only cos   0

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry r Angles in Quadrant II P 2 (  a, b) What are the coordinates of P 2 ?  AOP 2 = 180   

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry P 2 (  a, b)  AOP 2 = 180    Angles in Quadrant II r

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry P 3 (  a,  b) Reflex angle  AOP 3 = 180    Angles in Quadrant III What are the coordinates of P 3 ? r

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry P 3 (  a,  b) Angles in Quadrant III Reflex angle  AOP 3 = 180    r

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry P 4 (a,  b) Reflex angle  AOP 4 = 360    Angles in Quadrant IV What are the coordinates of P 4 ? r

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Angles in Quadrant IV P 4 (a,  b) Reflex angle  AOP 4 = 360    r

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Negative Angles sin(  )  sin(360   )   sin  cos(  )  cos(360   )  cos  tan(  )  tan(360   )   tan 

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Trigonometric Identities

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Graphs of Trigonometric Functions What are the values of the above trigonometric functions?

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Graphs of Trigonometric Functions

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Graph of Sine Function y  sin 

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Period of Sine Function y  sin  y  sin   1 11 O 360  Since the graph of sine function y  sin  will repeat once in every 360 , its period is 360 . sin(360n    )  sin  for any integer n. 360  180  180  360  540  720  900  1080 

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Graph of Cosine Function y  cos 

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Period of Cosine Function y  cos  y  cos  1 11 O 360  180  180  360  540  720  900  1080   360  Since the graph of cosine function y  cos  will repeat once in every 360 , its period is 360 . cos(360n    )  cos  for any integer n.

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Graph of Tangent Function y  tan 

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry O 360  180  180  360  540  720  900  1080   180  Period of Tangent Function y  tan 

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Since the graph of tangent function y  tan  will repeat once in every 180 , its period is 180 . tan(180n    )  tan  for any integer n. Period of Tangent Function y  tan 

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry Transformation of Trigonometric Graphs How do the values of A, m,  and B affect the graph of y  A sin m(x   )  B?

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry When A  1, the amplitude of the graph is larger than the original one (A  1). When 0  A  1, the amplitude of the graph is smaller than the original one (A  1). Transformation of Trigonometric Graphs

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry When B  0, the graph translates upwards. When B  0, the graph translates downwards. Transformation of Trigonometric Graphs

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry When m  1, the period of the graph decreases. When 0  m  1, the period of the graph increases. Transformation of Trigonometric Graphs

2004 Chung Tai Educational Press © Chapter Examples Quit Chapter 7 Trigonometry When   0 , the graph translates to the right. When   0 , the graph translates to the left. Transformation of Trigonometric Graphs

End