Trigonometry
0 to ± 2p
Trigonometric functions of negative arcs or angles
Trigonometric functions of complementary angles
Trigonometric functions of supplementary angles
Trigonometric functions of arcs from  0 to ± 2p
Trigonometric functions of negative arcs or angles
 We say that arcs x and x′ are opposite if x + x′ = 0  or  x′ = -x. Comparing the corresponding sides of the congruent right-angled triangles in the right figure, OPxP  and  OPxP′ OSxS1  and  OSxS1′ OSyS2  and  OSyS2′ follows that we can express trigonometric functions of a negative arc (-x) by corresponding function of opposite arc x, that is
Example:   Given trigonometric functions of negative angle, arc or number should be expressed by the same functions of the positive angle, arc or number.
a)  sin (-200°),      b)  cos (-14p/3),      c)  tan (-11),      d)  cot (-750°).
Solution:  First express the angle a in decimal degrees, i.e.
a)  sin (-200°) = - sin 200°
c)  tan (-11) = - tan11 = - tan (3p + 1.575222...) = - tan 1.575222...
d)  cot (-750°) = - cot 750° = - cot (2 · 360° + 30°) = - cot 30°.
Trigonometric functions of complementary angles
Two angles, x and p/2 - x which form the right angle, are said to be complementary.
Thus, comparing the corresponding sides of the congruent right-angled triangles in the below figure,
OPxP  and  OPxP,       OSxS1  and  OSyS2    and     OSyS2  and  OSxS1
Example:   The trigonometric functions of the given angle or arc should be expressed by corresponding function of the complementary angle.
a)  sin 30°,      b)  cos (p/2 - p/3),      c)  tan 1,      d)  cot 530°.
Solution:   a)  sin 30° = cos(90° - 30°) = cos 60°
d)  cot 530° = tan (90° - 530°) = tan (- 440°) = - tan 440° = - tan (2 · 180° + 80°)
= - tan 80° = - cot (90° - 80°) = - cot 10°.
 Example:   Simplify the expression
 Solution:
Trigonometric functions of supplementary angles
Two angles, x and p - x, which when added form a straight
angle, are said to be supplementary.
Comparing the corresponding sides of the congruent right-angled
triangles in the right figure,
 Px′P′ = PxP   => sin (p - x) = sin x
 OPx′ = -OPx   => cos (p - x) = - cos x
 SxS1′ = -SxS1   => tan (p - x) = - tan x
 SyS2′ = -SyS2   => cot (p - x) = - cot x

Example:   Trigonometric functions of a given arc, angle or number should be expressed by the corresponding function of the supplementary angle.
a)  sin 5p/6,      b)  cos (-320°),      c)  tan (p - 1),      d)  cot 30°.
Solution:   a)  sin 5p/6 = sin (p - 5p/6) = sin p/6
b)  cos (-320°) = cos (180° - 500°) = - cos 500° = - cos (360° + 140°)
= cos (180° - 140°) = cos 40°
c)  tan (p - 1) =  - tan 1
d)  cot 30° =  - cot (180° - 30°) =  - cot 150°.
Trigonometry contents A