9.0 trigonometry ii 2011
TRANSCRIPT
Chapter 9II9.1 Values of Sine , Cosine ( 00 3600 ) and Tangent
9.2 Graphs Sine, Cosine and Tangent
TRIGONOMETRIC RATIOOPPOSITE SIDE (O)
Sin
ADJACENT SIDE (A)
H = A Cos H Tan = O A
= O
9.1 Values of Sine , Cosine and Tangent ( 00 3600 ) Identify the quadrants and angles in the unit circle A unit circle is a circle of radius of 1 unit with its centre at the origin
The axes divide a unit circle into four equal parts, namely quadrants I, II, III and IV as shown in Diagram 1 Angles are measured from the x-axis in an anti-clockwise direction as shown in Diagram 2
y 1II Second quadrant I First quadrant
Diagram 1
-1III Third quadrant
0IV Fourth quadrant
1
x
-1
900
1
Diagram 2
II1800
I0 100
-1
III-12700
IV
Quadrant
LocationFrom positive x-axis to positive y-axis From positive y-axis to negative x-axis From positive x-axis to positive y-axis From positive y-axis to positive x-axis
I00 < < 900
II900 < < 1800
III1800 < < 2700
IV2700 < < 3600
Determine the values of the y-coordinate, x-coordinate and the ratio of the y-coordinate to the x-coordinate of several points on the circumference of a unit circle
ExampleBased on the unit circle (Diagram 3), complete Table 1
y
1 P Q Diagram 3x
-1
0
1
R -1
S
Point x-coordinate y-coordinate y-coordinate X-coordinate
P Q R S Table 1
y
10.7
P0.4 0.8 0.7
Q Diagram 3-0.8 -1 -0.9
0
1
x
R
-0.6
S
-1
SolutionPoint x-coordinate y-coordinate y-coordinate X-coordinate
P Q R S
0.7 -0.9 -0.8 0.8
0.7 0.4 -0.6 -0.6Table 1
0 .6 3 ! 0.8 4 0.6 3 ! 0.8 4
0.7 !1 0.7 0.4 4 ! 0.9 9
Verify that, for angle in quadrant 1, sin = y-coordinate, cos = x-coordinate and tan = y-coordinate x-coordinate
Quadrant 2
1
y
Quadrant1
P(x,y)1 Y
-1
0
x
1 x
Quadrant 3
-1
Quadrant 4
yQuadrant 2
1
Quadrant1
y P(x,y)
10
Y
Sin1 x
= =
Y
1Y
-1
Quadrant 3
-1
Quadrant 4
yQuadrant 2
1
Quadrant 1
x P(x,y)
10 x
-1
x Cos = 1 x =x 1
Quadrant 3
-1
Quadrant 4
yQuadrant 2
1
Quadrant 1
P(x,y) y x
Tan1 x
y = x
-1
0
Quadrant 3
-1
Quadrant 4
Quadrant 2
y 1
Quadrant 1 Sin
= y- coordinate P(x,y) of P Cos = x-coordinate of P x 1 Tan = y-coordinate of P x-coordinate of P
1
Y
-1
0
x
Quadrant 3
-1 Quadrant 4
Determine the values of sine, cosine and tangent of an angle in quadrant 1
ExampleBased on the quadrant of a unit circle (Diagram 4), determine the values of (a) sin 240 (b) cos 240 (c) tan 240
y
1
P(0.9,0.4)
Diagram 4
240
-1
0
1
x
-1
Solution
y
1
Sin 240 = y-coordinate of P = 0.4 P(0.9,0.4) 240 Cos 240 = x-coordinate of P = 0.9
-1
0
1
x
Tan 240 = y-coordinate of P x-coordinate of P = 0.4 0.9 = 0.44
-1
Q2
y 10.8
Sin 5308 = Q 1 y-coordinate P(x,y) of P = 0.8 Cos 5308= x-coordinate of P = 0.65308 0.6
-1
0
Q3
-1
Q4
x 1 Tan 5308 = y-coordinate of P x-coodinate of P = 1.3333
Determine the values of sine , cosine and tangent for ( 00 3600 )
ExampleBased on the unit circle (Diagram 5), determine the values of (a) sin 1270 (b) cos 1270 (c) tan 1270
y
1 P
Diagram 5
1270
-1
0
1
x
-1
SolutionP
y
10.8
Sin 1270 = y-coordinate of P = 0.8 Cos 1270 = x-coordinate of P = -0.6
1270
-1
-0.6
0
1
x
Tan 1270 = y-coordinate of P x-coordinate of P = 0.8 -0.6 = -1.33
-1
Determine whether the values of sine, cosine and tangent of an angle quadrants I,II,III and IV are positive or negative
in quadrant I ( 00