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16.1 POSI TIVE & NEG ATIV E AN GLES (a) Posi tiv e ang les – angles measured in the anticlockwise direction from the positive x-axis. (b) Neg ati ve ang les – a ngl es mea sur ed in the clockwise direction from the positive x-axis. Exercise 16.1 Represent each of the following angles in a unit circle. Then, state (i) the qua dra nt i n whic h t he a ngl es loc ate d, (ii) the corresponding acute angle. (a) 150 o (b) 315 o (c) 225 o (d)  6 (e) 3 2π   (f)  4 7π   16.2 (A) THE SIX TRIGONOMETRIC FUNCTIONS (i) sin θ = r  y if r = 1, then sin θ = y (ii) cos θ = r  x if r = 1, then cos θ = x  r  y (iii) tan θ =  x  y = θ θ cos sin   x (iv) cosec θ =  y r if r = 1, then cosec θ =  y 1 = θ sin 1 (v) sec θ =  x r if r = 1, then sec θ =  x 1 = θ cos 1 (vi) cot θ =  y  x = θ tan 1 = θ θ sin cos 16.2 (B) COMPLEMENT ANGLES (i) sin θ = cos (90 o  θ) (ii) cos θ = sin (90 o   θ) (i ii ) ta n θ = cot (90 o   θ) (i v) cosec θ = sec (90 o   θ) (v) sec θ = cosec (90 o   θ) θ

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16.1 POSITIVE & NEGATIVE ANGLES

(a) Positive angles angles measured in the

anticlockwisedirection

from the positive x-axis.

(b) Negative angles angles measured in the clockwise directionfrom the positive x-axis.

Exercise 16.1Represent each of the following angles in a unit circle. Then, state

(i) the quadrant in which the angles located,

(ii) the corresponding acute angle.

(a) 150o (b) 315o (c) 225o

(d) 6

(e)3

2

(f) 4

7

16.2 (A) THE SIX TRIGONOMETRIC FUNCTIONS(i) sin =

r

yif r= 1, then sin =y

(ii) cos =r

xif r = 1, then cos =x r y

(iii) tan =x

y=

cos

sin x

(iv) cosec =y

r if r = 1, then cosec =y

1 = sin1

(v) sec =x

rif r = 1, then sec =

x

1= cos

1

(vi) cot =y

x= tan

1=

sin

cos

16.2 (B) COMPLEMENT ANGLES

(i) sin = cos (90o )

(ii) cos = sin (90o)

(iii) tan = cot (90o)

(iv) cosec = sec (90o)

(v) sec = cosec (90o)

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(vi) cot = tan (90o)

16.2 (C) RELATIONSHIPS BETWEEN ANGLES > 90O

AND ITS ACUTE ANGLES

Quadrant II: Quadrant IV:sin = sin (180o) sin = sin (360o)

cos = cos (180o) cos = cos (360o)

tan = tan (180o) tan = tan (360o)

Quadrant III: Note:sin ( 180o) = sin If is the corresponding acute

cos ( 180o) = cos angle in the quadrant, then angle

tan ( 180o) = tan in Quadrant III is (180 + ).

16.2 (D) SPECIAL ANGLES: ( 0O, 30O, 45O, 60O, 90O, 180O, 270O, 360O)

0O

30O

45O

60O

90O

180O

270O

360O

sin 02

11 0 1 0

cos 12

10 1 0 1

tan 0 1 0 0

Exercise 16.2:

1. Given that sin =5

3, find the value of each of the following

a) cos

2

Tangent

positive(180o)

positive

( )

Cosinepositive

(360o)

Sinepositive

(180o)

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b) cosec

c) tan

d) cot

2. Given cos

=

p and 180

o