A. Trigonometric Ratios
of an Angle
Trigonometric Ratios of Right Triangle
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Create three right triangles with right angle length 3 cm and 4
cm, respectively name ABC, 6 cm and 8 cm, name DEF, and 9 cm
and 12 cm, name PQR.
R
F
C
A
B
D
E
P
Q
Using Pythagorean theorem, you should be able to calculate
length of side AC, DF, and PR (hypotenuse) of 15 cm, 10 cm, and
15 cm, respectively. Further, compare A, D, and P measure. In
fact, A=B=P.
Trigonometric ratios for A in right triangle ABC is defined as
follows.
a. BC/AC is called sine A abbreviated by sin A
b. AB/AC is called cosine A abbreviated by cos A
c. BC/AB is called tangent A abbreviated by tan A
d. AB/BC is called cotangent A abbrevaited by cot A
e. AC/AB is called secant A abbreviated by sec A
f. AC/BC is called cosecant A abbreviated by csc A
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At whole, trigonometric ratios values of angle
in the three
triangles are as follows.
sin
= 3/5
d. cot alpha
=4/3
cos alpha
= 4/5
e. sec alpha
=5/4
tan alpha
= 3/4
f. csc alpha
=4/3
Of those ratios, following relationship are obtained.
Trigonometric Ratios on Coordinates
System
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Observe beside figure. Beside figure is a circle with centre
O(0,0) and radius r. Angle
is angle between positive
X-axis and line OP. Line OP can be rotated that measure of
angle
ranges between
Coordinates of P is P(x,y).Length of circle radius is r so that
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Trigonometric Ratios for Angle
are definited as follows
if angle = 0o, then the point coordinates of P(x,y) is P(r,0).
It means that as follows
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Sin α = y/r
Cot α = x/y
tan α = y/x
cot α = x/y
sec α = r/x
csc α = r/y
If angle α = 0o, the the point coordinate of P(x,y) is P(r,0).
It means that as follows.
sin 0o = y/r = 0/r = 0; cos 0o = x/r = r/r = 1; tan 0o = y/x
= 0/r = 0
if angle α = 90o, then coordinate of point P(x,y) is P(0,r). It
means that as follows.
sin 90o = y/x = r/r = 1; cos 90o = x/r = 0/r; tan 90o = y/x
= r/0 (undefined).
Extraordianary Angle Trigonometric Ratios
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There are some angles that is trigonometric ratios value
can be determined without trigonometric table or calculator
aid, for examples 0o, 30o, 45o, 60o, and 90o. Those angles
are caled extraordianary angles or special angles, which its
trigonometric ratio value can can be determined among
orthers, by using trigonometric ratios definition on the
circle at centre point o(0,0) and radius r. Subject on 0o and
90o have been studied before.
a. Trigonometric Ratios Value of Angle 45o.
Look out at beside figure.
It
seems that XOY is right
angle.
Hence, if angle XOP
= 45o,
then angle YOP =
45o.
1) sin 45o = y/r = (1/2r√2) / r = 1/2√2
2) cos 45o = x/r = (1/2r√2) / r = 1/2√2
3) tan 45o = y/x = (1/2r√2) / (1/2r√2)=1
b. Trigonometric ratios values of angle 30o and 60o
Look at biside figure. F line
PQ
is perpendicular to OX and
angle XOP = 30 o, then
triangle OPQ = 60o.
Because
OP = OQ, then
triangle OPQ is
equalteral so
that length OP
=OQ = PQ.
Point R is
intersection of
PQ and OX so
that PR = RQ
= 1/2r.
hence,it is easly shown
sin 30o = ½
cos 30o = 1/2√3
tan 30o = 1/3√3
that.
sin 60o = 1/2√3
cos 60o = ½
tan 60o = √3
extraordinary angles trigonometric ratios value fo sin α, csc α, and
tan α are completely summarized in the following table.
Other extraordinary angles trigonometric ratios values, namely
sec α, csc α, and cot α can be obtained by using the inverse formula.
Extraordinary Angles
Trigonometr
ic ratio
0o
30o
45o
60o
90o
Sin α
0
1/2
1/2√2
1/2√3
1
Cos α
1
1/2√3
1/2√2
1/2
0
Tan α
0
1/3√3
1
√3
undefined
II
I
Coordinate pivots at beside figure divides
coordinate plane into for areas, which
furhtermore, is called quardant.
Thus, angle
quantity can be classified
into four quadrants, III
IV
namely as follows.
a. quadrant I : 0o < α ≤90o
b. quadrant II : 90o < α ≤ 180o
c. quadrant III: 180o < α ≤ 270o
d. quadrant IV: 270o < α ≤ 360o
If α is at quadrant I, then x and y are positve.
sin α = y/r > 0
cos α = x/r > 0
tan α = y/x > 0
Determined Uknow Right-Angled Triangle
Component
Trigonometric value can be used to determined side length
of right-angled triangle if of its acute angle is know. In
addidion, we caan also determined the quantity of two
acute angles if at least there are two sides known.
Formula of Related Angled Trigonometric Ratios
Related angle is angle pairs that have any relation so that
trigonometric ratios of its angles fulfill certain formula.
Some related angles trigonometic ratios formula will b
explainned in the following section.
a. Angle α with 90o – α
Point P’(x’,y’) is a reflection result of point P9x,y) towards
line y = x so that x’ = y and y’ = x. Therefore, coordinate
of point P’(x’,y’) is p’(y,x).
<xop = α
<xop = 90o – α
OP = OP’= r
Considering coordinate of point P’(x’.y’) =nP’(y,x), then for
angle 90o – α the following apply.
sin (90o – α) = y’/OP’
tan (90o – α) =
y’/x’
= x/r = cos α
= x/y = cot α
cos (90o – α) = x’/OP’
= y/r = sin α
Determined the values cot, sec, and csc of angle 90o – α.
b. Angle α with 180o – α
Line segment OP’ on beside figure is shadow of line
segment OP if ppoint P is resflected aggaint Y-axis o that
the shadow is P’. Because o that reflections, the following
relations accur.
a. <XOP = α; <XOP’ = (180o – α)
b. r’ = r’,x’ = -x;y’ = y.
thus, coordinate of point P’(x’,y’) = P’(-x,y).
brcause in coordinate P’(x’,y’) has a relation P’(x’,y’) = (x,y), for angle (180o – α) the following trigonometric ratios
is obtained.
a. sin (180o – α) = y/r = sin α
b. cos (180o – α) = -x/r = -cos α
c. tan (180o – α) = y/-x = -tan α
determine value cot, sec, and csc of angle 180o – α.
c. angle α with (180o + α)
Point P” in the following figure is the shadow from point
P after passing twice reflection respectively, namely against
Y-axis is continued to X-axis. Thus, line segment OP is also
shadow of line segment OP through twice reflection in
similar way that is against Y-axis is continued to X-axis.
Because of the reflection , the following relation is
obtained.
a. <XOP = α; <XOP” = (180o = α)
b. r” = r, x” = -x; y” = -y
therefore, P”(x”,y”) = P(-x,-y).
As a consequence of the relation, for angle (180o + α),
the following trigonometric ratios is found.
a. sin (180o + α) = -y/r = - y/r = -sin α
b. cos (180o + α) = -x/r = - x/r = -cos α
c. tan (180o + α) = -y/-x = y/x = tan α
Determine the values cot, sec, and csc of angle 180o + α.
d. angle α with (360o – α)
Point P is reflected against X-axis so that image is point P’.
Thus, the length of line segment OP equals to line segment OP.
Due to the reflection, the following relation is obtained.
a. <OXP = α; <XOP’ =m(360o - α ) = - α
b. r’ = r; x’ = x; y’ = -y
Therefore, P’(x’,y’) = P(x,-y)
As a conequence of this relation, trigonometric ratios for
(360o – α) angle is as follows.
a. sin (360o - α ) = -y/r = -sin α
b. cos (360o - α ) = x/r = cos α
c. tan (360+ - α ) = -y/x = -tan α
Determine cot,sec, and csc value for angle 360o - α .
Angle (360o - α ) may be also viewed as angle - α so that
trigonometric ratios valu for angle - α is as follows.
sin (- α ) = -sin α
cot (- α ) = -cot α
cos (i α ) = cos α
sec (- α ) = sec α
tan (- α ) = -tan α
csc (- α ) = - csc α
e. angle α with (k x 360o + α)
if line OP on beside figure is rotated in one full rotation
as much as k times and P’(x’,y’) is rotation resulted from
P(x,y), then P’(x’,y’) will be very coincide with P(x.y) so
that x’ = x,y’ = y is obtained, and length OP’ = length OP =
r.
By paying attention to pervious result, that is tan
(180+ α) = tan α and α (180+ α) = cot v then if k is
integer, applies as follows.
sin (k x 360o + α) = sin α
cos (k x 360o + α) = cos α
tan (k x 180o + α) = tan α
sec (k x 360o + α) = sec α
csc (k x 360o + α) = csc α
cot (k x 180o + α) = cot α