Trigonometric waveforms 135
curve appears to be discontinuous and repeats at
intervals of 180 ◦ .
14.2 Angles of any magnitude
(i) Figure 14.2 shows rectangular axes XX’ and YY’
intersecting at origin 0. As with graphical work,
measurements made to the right and above 0 are
positive while those to the left and downwards
are negative. Let OA be free to rotate about 0.
By convention, when OA moves anticlockwise
angular measurement is considered positive, and
vice-versa.
908
3608
2708
1808
X 9
X
Y 9
Y
08
A
Quadrant 2
Quadrant 3
Quadrant 4
Quadrant 1
0
2
1
1
2
1
2
Figure 14.2
(ii) Let OA be rotated anticlockwise so that θ 1 is any
angle in the first quadrant and let perpendicular
AB be constructed to form the right-angled triangle OAB (see Fig. 14.3). Since all three sides
of the triangle are positive, all six trigonometric
ratios are positive in the first quadrant. (Note: OA
is always positive since it is the radius of a circle.)
1808
908
2708
3608
08
2
3
4
1
Quadrant 2
Quadrant 3
Quadrant 4
Quadrant 1
0
A
B
A
A
C
D
E
A
2
2
2
1
1
1
1
1
1
1
Figure 14.3
(iii) Let OA be further rotated so that θ 2 is any angle
in the second quadrant and let AC be constructed
to form the right-angled triangle OAC. Then:
sin θ 2 =
+
+
= +
cos θ 2 =
−
+
= −
tan θ 2 =
+
−
= −
cosec θ 2 =
+
+
= +
sec θ 2 =
+
−
= −
cot θ 2 =
−
+
= −
(iv) Let OA be further rotated so that θ 3 is any angle
in the third quadrant and let AD be constructed
to form the right-angled triangle OAD. Then:
sin θ 3 =
−
+
= − (and hence cosec θ 3 is −)
cos θ 3 =
−
+
= − (and hence sec θ 3 is +)
tan θ 3 =
−
−
= + (and hence cot θ 3 is −)
(v) Let OA be further rotated so that θ 4 is any angle
in the fourth quadrant and let AE be constructed
to form the right-angled triangle OAE. Then:
sin θ 4 =
−
+
= − (and hence cosec θ 4 is −)
cos θ 4 =
+
+
= + (and hence sec θ 4 is +)
tan θ 4 =
−
+
= − (and hence cot θ 4 is −)
(vi) The results obtained in (ii) to (v) are summarized
in Fig. 14.4. The letters underlined spell the word
CAST when starting in the fourth quadrant and
moving in an anticlockwise direction.
908
1808
2708
3608
08
Sine (and cosecant)
positive
Tangent
(and cotangent)
positive
Cosine
(and secant)
positive
All positive
Figure 14.4
curve appears to be discontinuous and repeats at
intervals of 180 ◦ .
14.2 Angles of any magnitude
(i) Figure 14.2 shows rectangular axes XX’ and YY’
intersecting at origin 0. As with graphical work,
measurements made to the right and above 0 are
positive while those to the left and downwards
are negative. Let OA be free to rotate about 0.
By convention, when OA moves anticlockwise
angular measurement is considered positive, and
vice-versa.
908
3608
2708
1808
X 9
X
Y 9
Y
08
A
Quadrant 2
Quadrant 3
Quadrant 4
Quadrant 1
0
2
1
1
2
1
2
Figure 14.2
(ii) Let OA be rotated anticlockwise so that θ 1 is any
angle in the first quadrant and let perpendicular
AB be constructed to form the right-angled triangle OAB (see Fig. 14.3). Since all three sides
of the triangle are positive, all six trigonometric
ratios are positive in the first quadrant. (Note: OA
is always positive since it is the radius of a circle.)
1808
908
2708
3608
08
2
3
4
1
Quadrant 2
Quadrant 3
Quadrant 4
Quadrant 1
0
A
B
A
A
C
D
E
A
2
2
2
1
1
1
1
1
1
1
Figure 14.3
(iii) Let OA be further rotated so that θ 2 is any angle
in the second quadrant and let AC be constructed
to form the right-angled triangle OAC. Then:
sin θ 2 =
+
+
= +
cos θ 2 =
−
+
= −
tan θ 2 =
+
−
= −
cosec θ 2 =
+
+
= +
sec θ 2 =
+
−
= −
cot θ 2 =
−
+
= −
(iv) Let OA be further rotated so that θ 3 is any angle
in the third quadrant and let AD be constructed
to form the right-angled triangle OAD. Then:
sin θ 3 =
−
+
= − (and hence cosec θ 3 is −)
cos θ 3 =
−
+
= − (and hence sec θ 3 is +)
tan θ 3 =
−
−
= + (and hence cot θ 3 is −)
(v) Let OA be further rotated so that θ 4 is any angle
in the fourth quadrant and let AE be constructed
to form the right-angled triangle OAE. Then:
sin θ 4 =
−
+
= − (and hence cosec θ 4 is −)
cos θ 4 =
+
+
= + (and hence sec θ 4 is +)
tan θ 4 =
−
+
= − (and hence cot θ 4 is −)
(vi) The results obtained in (ii) to (v) are summarized
in Fig. 14.4. The letters underlined spell the word
CAST when starting in the fourth quadrant and
moving in an anticlockwise direction.
908
1808
2708
3608
08
Sine (and cosecant)
positive
Tangent
(and cotangent)
positive
Cosine
(and secant)
positive
All positive
Figure 14.4