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.
Example:   Calculate,  sin 3p/2 · cos(- p) + tan 5p/4.
Solution:  sin 3p/2 · cos(- p) + tan 5p/4 = - 1 · (- 1) + tan (p + p/4) = 1 + tan p/4 = 1 + 1 = 2.
 Example:   Calculate,
Solution:
Example:  Prove the identity,
cos2 p/3 · sin (p/2 - x) - cos (p - x) · cos2 p/6 = tan (p/2 + x) · sin (2p - x).
Solution:  Since  sin (p/2 - x) = cos x,   cos (p - x) = - cos x,   tan (p/2 + x) = - cot x
and  sin (2p - x) = - sin x  then,
(cos p/3)2 · cos x - (- cos x) · (cos p/6)2 = - cot x · (- sin x),
(1/2)2 · cos x + (Ö3/2)2 · cos x = (cos x/sin x) · sin x  =>  cos x = cos x.
Example:  Prove the identity,
cot2 (p + x) · cos2 (p/2 + x) + sin (- x) · sin (p + x) = tan (2p - x) · cot (- x).
Solution:    [cot (p + x)]2 · [cos (p/2 + x)]2 + (- sin x) · sin (p + x) = (- tan x) · (- cot x),
(cot x)2 · (- sin x)2 + (- sin x) · (- sin x) = (sin x/cos x) · (cos x/sin x)
cos2 x + sin2 x = (sin x/cos x) · (cos x/sin x) = 1.
Pre-calculus contents C