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