Trigonometry
0 to ± 2p
p/2
p
Trigonometric functions of arcs whose sum is 2p
Trigonometric functions of angles lying on axes
Trigonometric functions of arcs that differ on p/2
Comparing the corresponding sides of the congruent right-angled triangles, in the right figure, we get the relations of trigonometric functions of an arc x and the arc p/2 + x
 Px′P′ = OPx   => sin (p/2 + x) = cos x
 OPx′ = -PxP   => cos (p/2 + x) = -sin x
 SxS1′ = -SyS2  => tan (p/2 + x) = -cot x
 SyS2′ = -SxS1  => cot (p/2 + x) = -tan x
Example:   Trigonometric functions of a given arc, angle or number should be expressed by the corresponding function of angle which differ from the given for 90° (p/2).
a)  sin 1,      b)  cos 150°,      c)  tan (-7p/4),      d)  cot 50°.
Solution:   a)  sin 1 = - cos (p/2 + 1) =  - cos 2.570796...
b)  cos 150° = cos (90° + 60°) = - sin 60°
c)  tan (-7p/4) =  - cot (p/2 - 7p/4) =  - cot ( - 5p/4) = cot (p + p/4) = cot p/4
d)  cot 50° =  - tan (90° + 50°) = - tan 140°.
Trigonometric functions of arcs that differ on p
Comparing the corresponding sides of the congruent right-angled
triangles, in the right figure, we get the relations of trigonometric
functions of an arc x and the arc p + x
 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 angle which differ from the given for 180° (p).
a)  sin 235°,      b)  cos p/6,      c)  tan (-300°),      d)  cot 4.
Solution:   a)  sin 235° = sin (180° + 55°) = - sin 55°
b)  cos p/6 = - cos (p + p/6) = - cos 7p/6
c)  tan (-300°) =  tan (180° - 300°) =  tan (-120°) = - tan 120°
d)  cot 4 = cot (p + 0.858407...) = cot 0.858407....
Example:   Simplify expression  cot (p - x) · cos (p/2 + x) + tan (p/2 - x) · tan (p + x) - cos (- x)
Solution:   cot (p - x) · cos (p/2 + x) + tan (p/2 - x) · tan (p + x) - cos (- x) =
= - cot x · ( - sin x) + cot x · tan x - cos x = cos x + 1 - cos x = 1.
Trigonometric functions of arcs whose sum is 2p
The right figure shows relations between sides of the congruent
right-angled triangles as follows,
 PxP′ =  - PxP   => sin (2p - x) = - sin x
 OPx = cos (2p - x) = cos x cos (2p - x) = cos x
 SxS1′ = - SxS1   => tan (2p - x) = - tan x
 SyS2′ = - SyS2   => cot (2p - x) = - cot x
Example:   Trigonometric functions of a given arc, angle or number should be expressed by the corresponding function of angle which when added with a given make 360° (2p).
a)  sin p/3,      b)  cos 1,      c)  tan 330°,      d)  cot 10p/11.
Solution:   a)  sin p/3 = -sin (2p - p/3) = - sin 5p/3
b)  cos 1 = - cos (2p - 1) = cos 5.283185...
c)  tan 330° =  tan (360° - 330°) = - tan 30°
d)  cot 10p/11 = - cot (2p - 10p/11) =  - cot p/11.
Example:   Prove that  sin 320° + cos 50° = 0.
Solution:   Since      sin 320° = sin (360° - 40°) = - sin 40°,    and as    cos 50° = sin 40°
then      - sin 40° + sin 40° = 0.
Trigonometric functions of angles lying on axes
Values of trigonometric functions of characteristic arcs, 0, p/2, 3p/2 and 2p follow directly from the definitions.
Thus, for functions sine and cosine from the above figure we read the coordinates of the arc terminal point P that is, for the sine function we read the ordinate while for the cosine function the abscissa of the terminal point.
Therefore,         sin 0 = 0,          sin p/2 = 1,          sin p = 0,          sin 3p/2 = -1,          sin 2p = 0,
and         cos 0 = 1,         cos p/2 = 0,        cos p = -1,        cos 3p/2 = 0,          cos 2p = 1.
Point S1 whose ordinate determines the value of the function tangent, for the arcs, 0, p and 2p, coincide with the initial point P1 of the arc, i.e., lies on the x-axis, see the down figure.
Thus,        tan 0 = 0,          tan p = 0,          tan 2p = 0,
while for arcs, p/2 and 3p/2  their terminal side or its extension lies on the y-axis, that is parallel with tangent x = 1. There is no intersection S1 and we say that for these arcs the function tangent is undefined.
However, if we follow the intersection point S1 while the arc increases from 0 to p/2 we see that it moves away the x-axis and its ordinate tan a1  tends to infinity (+ oo) which can be written as,
when  a1 ® p/2,   tan a1 ®  + oo   or    tan p/2 = oo .
If we continue to follow changes of the values of the function tangent, i.e., changes of the ordinates of the intersection S1 while the arc increases from p/2 to p that is, while the terminal side of the angle a2 or its extension continue rotates in the positive direction, we see that point S1 moves toward the x-axis and at the same time its ordinate tan a2 increases from  - oo  to 0.
Thus we can write,   tan p/2 = ± oo .         Examining the same way it follows that,   tan 3p/2 = ± oo .
The intersection point S2, whose abscissas determine the values of the function cotangent, coincide with the point Pp/2 for the arcs p/2 and 3p/2 on the y-axis, so
cot p/2 = 0   and   cot 3p/2 = 0
while for arcs, 0(2p) and p, the terminal side of the corresponding central angle, or its extension, lies on the x-axis parallel with the tangent y = 1, so there is no intersection point.
We say that the function cotangent is undefined for those arcs.
To determine bounds of the values that the function cotangent takes while the terminal point of an arc rounds the unit circle in the positive direction passing through mentioned characteristic values, 0(2p) and p, we should follow the intersection point S2 on the tangent y = 1, i.e., the changes of its abscissas cot a, see the above figure. Thus,
Functions contents B