A. Trigonometric Ratios A. Trigonometric Ratios of an Angleof an Angle

Trigonometric Ratios of Right TriangleTrigonometric Ratios of Right Triangle Create three right triangles with right angle length 3 cm and 4 cm, 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 respectively name ABC, 6 cm and 8 cm, name DEF, and 9 cm and 12 cm, name PQR. R12 cm, name PQR. R

FF CC

A B D E P QA B D E P Q Using Pythagorean theorem, you should be able to calculate Using Pythagorean theorem, you should be able to calculate

length of side AC, DF, and PR (hypotenuse) of 15 cm, 10 cm, and 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 15 cm, respectively. Further, compare A, D, and P measure. In fact, A=B=P.fact, A=B=P.

Trigonometric ratios for A in right triangle ABC is defined as Trigonometric ratios for A in right triangle ABC is defined as follows.follows.a. BC/AC is called sine A abbreviated by sin Aa. BC/AC is called sine A abbreviated by sin Ab. AB/AC is called cosine A abbreviated by cos Ab. AB/AC is called cosine A abbreviated by cos Ac. BC/AB is called tangent A abbreviated by tan Ac. BC/AB is called tangent A abbreviated by tan Ad. AB/BC is called cotangent A abbrevaited by cot Ad. AB/BC is called cotangent A abbrevaited by cot Ae. AC/AB is called secant A abbreviated by sec Ae. AC/AB is called secant A abbreviated by sec Af. AC/BC is called cosecant A abbreviated by csc Af. AC/BC is called cosecant A abbreviated by csc A

At whole, trigonometric ratios values of angle in the three At whole, trigonometric ratios values of angle in the three triangles are as follows.triangles are as follows.

sin = 3/5sin = 3/5 d. cot alphad. cot alpha =4/3=4/3

cos alphacos alpha = 4/5= 4/5 e. sec alphae. sec alpha =5/4=5/4

tan alphatan alpha = 3/4= 3/4 f. csc alphaf. csc alpha =4/3=4/3 Of those ratios, following relationship are obtained.Of those ratios, following relationship are obtained.

Trigonometric Ratios on Coordinates Trigonometric Ratios on Coordinates SystemSystem

Observe beside figure. Beside figure is a circle with centre Observe beside figure. Beside figure is a circle with centre O(0,0)O(0,0) and radius and radius r. r. Angle Angle is angle between positive X-is angle between positive X-axis and line OP. Line OP can be rotated that measure of axis and line OP. Line OP can be rotated that measure of angleangle ranges between ranges between

Coordinates of P is P(x,y).Length of circle radius is r so thatCoordinates of P is P(x,y).Length of circle radius is r so that

Trigonometric Ratios for AngleTrigonometric Ratios for Angle

are definited as followsare definited as follows

if angle = 0if angle = 0o,o, then the point coordinates of P(x,y) is P(r,0). then the point coordinates of P(x,y) is P(r,0). It means that as followsIt means that as follows

Sin Sin αα = y/r = y/r cot cot αα = x/y = x/y Cot Cot αα = x/y = x/y sec sec αα = r/x = r/x tan tan αα = y/x = y/x csc csc αα = r/y = r/y

If angle αIf angle α = 0= 0o, o, the the point coordinate of P(x,y) is P(r,0). It the the point coordinate of P(x,y) is P(r,0). It means that as follows.means that as follows.

sin 0sin 0oo = y/r = 0/r = 0; cos 0 = y/r = 0/r = 0; cos 0oo = x/r = r/r = 1; tan 0 = x/r = r/r = 1; tan 0oo = y/x = = y/x = 0/r = 00/r = 0

if angle α = 90if angle α = 90oo, then coordinate of point P(x,y) is P(0,r). It , then coordinate of point P(x,y) is P(0,r). It means that as follows.means that as follows.

sin 90sin 90oo = y/x = r/r = 1; cos 90 = y/x = r/r = 1; cos 90oo = x/r = 0/r; tan 90 = x/r = 0/r; tan 90oo = y/x = = y/x = r/0 (undefined).r/0 (undefined).

Extraordianary Angle Trigonometric RatiosExtraordianary Angle Trigonometric Ratios There are some angles that is trigonometric ratios value There are some angles that is trigonometric ratios value

can be determined without trigonometric table or calculator can be determined without trigonometric table or calculator aid, for examples 0aid, for examples 0oo, 30, 30oo, 45, 45oo, 60, 60oo, and 90, and 90oo. Those angles . Those angles are caled extraordianary angles or special angles, which its are caled extraordianary angles or special angles, which its trigonometric ratio value can can be determined among trigonometric ratio value can can be determined among orthers, by using trigonometric ratios definition on the circle orthers, by using trigonometric ratios definition on the circle at centre point o(0,0) and radius r. Subject on 0at centre point o(0,0) and radius r. Subject on 0oo and 90 and 90o o

have been studied before. have been studied before.

a. Trigonometric Ratios Value of Angle 45a. Trigonometric Ratios Value of Angle 45oo..

Look out at beside Look out at beside figure. It figure. It seems that XOY seems that XOY is right angle. is right angle. Hence, if Hence, if angle XOP = 45angle XOP = 45oo, , then then angle YOP = 45angle YOP = 45oo..

1) sin 451) sin 45oo = y/r = (1/2r√2) / r = 1/2√2 = y/r = (1/2r√2) / r = 1/2√22) cos 452) cos 45oo = x/r = (1/2r√2) / r = 1/2√2 = x/r = (1/2r√2) / r = 1/2√23) tan 453) tan 45oo = y/x = (1/2r√2) / (1/2r√2)=1 = y/x = (1/2r√2) / (1/2r√2)=1

b. Trigonometric ratios values of angle 30b. Trigonometric ratios values of angle 30oo and 60 and 60oo

Look at biside figure. F line Look at biside figure. F line PQ PQ is perpendicular to OX and is perpendicular to OX and

angle XOP = 30 o, then angle XOP = 30 o, then triangle OPQ = triangle OPQ =

60o. Because 60o. Because OP = OQ, then OP = OQ, then triangle OPQ is triangle OPQ is equalteral so that equalteral so that length OP length OP =OQ = PQ. Point R =OQ = PQ. Point R is is intersection of PQ intersection of PQ and OX so and OX so that PR = RQ = that PR = RQ = 1/2r.1/2r.

hence,it is easly shown that.hence,it is easly shown that.sin 30sin 30oo = ½ = ½ sin 60sin 60oo = 1/2√3 = 1/2√3cos 30cos 30oo = 1/2√3 = 1/2√3cos 60cos 60oo = ½ = ½tan 30tan 30oo = 1/3√3 = 1/3√3tan 60tan 60oo = √3 = √3

extraordinary angles trigonometric ratios value fo sin extraordinary angles trigonometric ratios value fo sin αα, csc , csc αα, , and tan and tan αα are completely summarized in the following table. are completely summarized in the following table.

Other extraordinary angles trigonometric ratios values, namely Other extraordinary angles trigonometric ratios values, namely sec sec αα, csc , csc αα, and cot , and cot αα can be obtained by using the inverse formula. can be obtained by using the inverse formula.

TrigonometrTrigonometric ratioic ratio

Extraordinary AnglesExtraordinary Angles

00oo 3030oo 4545oo 6060oo 9090oo

Sin Sin αα 00 1/21/2 1/2√21/2√2 1/2√31/2√3 11

Cos Cos αα 11 1/2√31/2√3 1/2√21/2√2 1/21/2 00

Tan Tan αα 00 1/3√31/3√3 11 √√33 undefinedundefined

IIII I I Coordinate pivots at beside figure Coordinate pivots at beside figure divides divides coordinate plane into for areas, coordinate plane into for areas, which which furhtermore, is called furhtermore, is called quardant. Thus, angle quardant. Thus, angle quantity can be quantity can be classified into four quadrants, III IVclassified into four quadrants, III IV namely namely as follows.as follows.

a. quadrant Ia. quadrant I : 0: 0oo < < αα ≤≤9090oo

b. quadrant IIb. quadrant II : 90: 90oo < < αα ≤≤ 180 180oo

c. quadrant III: 180c. quadrant III: 180oo < < αα ≤≤ 270 270oo

d. quadrant IV: 270d. quadrant IV: 270oo < < αα ≤≤ 360 360oo

If If αα is at quadrant I, then x and y are positve. is at quadrant I, then x and y are positve.

sin sin αα = y/r > 0 = y/r > 0

cos cos αα = x/r > 0 = x/r > 0

tan tan αα = y/x > 0 = y/x > 0

Determined Uknow Right-Angled Triangle Determined Uknow Right-Angled Triangle ComponentComponent

Trigonometric value can be used to determined side length Trigonometric value can be used to determined side length of right-angled triangle if of its acute angle is know. In of right-angled triangle if of its acute angle is know. In addidion, we caan also determined the quantity of two addidion, we caan also determined the quantity of two acute angles if at least there are two sides known.acute angles if at least there are two sides known.

Formula of Related Angled Trigonometric RatiosFormula of Related Angled Trigonometric RatiosRelated angle is angle pairs that have any relation so that Related angle is angle pairs that have any relation so that trigonometric ratios of its angles fulfill certain formula. trigonometric ratios of its angles fulfill certain formula. Some related angles trigonometic ratios formula will b Some related angles trigonometic ratios formula will b explainned in the following section.explainned in the following section.a. Angle a. Angle αα with 90 with 90oo – – ααPoint P’(x’,y’) is a reflection result of point P9x,y) towards 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 line y = x so that x’ = y and y’ = x. Therefore, coordinate of point P’(x’,y’) is p’(y,x).point P’(x’,y’) is p’(y,x).<xop = <xop = αα<xop = 90<xop = 90oo – – ααOP = OP’= rOP = OP’= rConsidering coordinate of point P’(x’.y’) =nP’(y,x), then for Considering coordinate of point P’(x’.y’) =nP’(y,x), then for angle 90angle 90oo – – αα the following apply. the following apply.

sin (90sin (90oo – – αα) = y’/OP’) = y’/OP’ tan (90tan (90oo – – αα) = y’/x’) = y’/x’= x/r = cos = x/r = cos αα = x/y = cot = x/y = cot

ααcos (90cos (90oo – – αα) = x’/OP’) = x’/OP’

= y/r = sin = y/r = sin ααDetermined the values cot, sec, and csc of angle 90Determined the values cot, sec, and csc of angle 90oo – – αα..

b. Angle b. Angle αα with 180 with 180oo – – αα

Line segment OP’ on beside figure is shadow of line Line segment OP’ on beside figure is shadow of line segment OP if ppoint P is resflected aggaint Y-axis o that segment OP if ppoint P is resflected aggaint Y-axis o that the shadow is P’. Because o that reflections, the following the shadow is P’. Because o that reflections, the following relations accur.relations accur.

a. <XOP = a. <XOP = αα; <XOP’ = (180; <XOP’ = (180oo – – αα))

b. r’ = r’,x’ = -x;y’ = y.b. r’ = r’,x’ = -x;y’ = y.

thus, coordinate of point P’(x’,y’) = P’(-x,y).thus, coordinate of point P’(x’,y’) = P’(-x,y).

brcause in coordinate P’(x’,y’) has a relation P’(x’,y’) = (-brcause in coordinate P’(x’,y’) has a relation P’(x’,y’) = (-x,y), for angle (180x,y), for angle (180oo – – αα) the following trigonometric ratios is ) the following trigonometric ratios is obtained.obtained.

a. sin (180a. sin (180oo – – αα) = y/r = sin ) = y/r = sin αα

b. cos (180b. cos (180oo – – αα) = -x/r = -cos ) = -x/r = -cos αα

c. tan (180c. tan (180o o – – αα) = y/-x = -tan ) = y/-x = -tan αα

determine value cot, sec, and csc of angle 180determine value cot, sec, and csc of angle 180oo – – αα..

c. angle c. angle αα with (180 with (180o o + + αα))

Point P” in the following figure is the shadow from Point P” in the following figure is the shadow from point P after passing twice reflection respectively, namely point P after passing twice reflection respectively, namely against Y-axis is continued to X-axis. Thus, line segment OP against Y-axis is continued to X-axis. Thus, line segment OP is also shadow of line segment OP through twice reflection is also shadow of line segment OP through twice reflection in similar way that is against Y-axis is continued to X-axis.in similar way that is against Y-axis is continued to X-axis.

Because of the reflection , the following relation is Because of the reflection , the following relation is obtained.obtained.

a. <XOP = a. <XOP = αα; <XOP” = (180; <XOP” = (180o o = = αα))

b. r” = r, x” = -x; y” = -yb. r” = r, x” = -x; y” = -y

therefore, P”(x”,y”) = P(-x,-y).therefore, P”(x”,y”) = P(-x,-y).

As a consequence of the relation, for angle (180As a consequence of the relation, for angle (180oo + + αα), the following trigonometric ratios is found.), the following trigonometric ratios is found.

a. sin (180a. sin (180oo + + αα) = -y/r = - y/r = -sin ) = -y/r = - y/r = -sin αα

b. cos (180b. cos (180oo + + αα) = -x/r = - x/r = -cos ) = -x/r = - x/r = -cos αα

c. tan (180c. tan (180oo + + αα) = -y/-x = y/x = tan ) = -y/-x = y/x = tan αα

Determine the values cot, sec, and csc of angle 180Determine the values cot, sec, and csc of angle 180oo + + αα..

d. angle d. angle αα with (360 with (360oo – – αα))

Point P is reflected against X-axis so that image is point P’. 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. Thus, the length of line segment OP equals to line segment OP. Due to the reflection, the following relation is obtained.Due to the reflection, the following relation is obtained.

a. <OXP = a. <OXP = αα; <XOP’; <XOP’ =m(360=m(360oo - - αα ) = - ) = - αα

b. r’ = r; x’ = x; y’ = -yb. r’ = r; x’ = x; y’ = -y

Therefore, P’(x’,y’) = P(x,-y)Therefore, P’(x’,y’) = P(x,-y)

As a conequence of this relation, trigonometric ratios for As a conequence of this relation, trigonometric ratios for (360(360oo – – αα) angle is as follows.) angle is as follows.

a. sin (360a. sin (360o o - - αα ) = -y/r = -sin ) = -y/r = -sin αα

b. cos (360b. cos (360oo - - αα ) = x/r = cos ) = x/r = cos αα

c. tan (360+ - c. tan (360+ - αα ) = -y/x = -tan ) = -y/x = -tan αα

Determine cot,sec, and csc value for angle 360Determine cot,sec, and csc value for angle 360oo - - αα . .

Angle (360Angle (360oo - - αα ) may be also viewed as angle - ) may be also viewed as angle - αα so that so that trigonometric ratios valu for angle - trigonometric ratios valu for angle - αα is as follows. is as follows.

sin (- sin (- αα ) = -sin ) = -sin αα cot (- cot (- αα ) = -cot ) = -cot αα

cos (i cos (i αα ) = cos ) = cos αα sec (- sec (- αα ) = sec ) = sec αα

tan (- tan (- αα ) = -tan ) = -tan αα csc (- csc (- αα ) = - csc ) = - csc αα

e. angle e. angle αα with (k x 360 with (k x 360oo + + αα))

if line OP on beside figure is rotated in one full if line OP on beside figure is rotated in one full rotation as much as k times and P’(x’,y’) is rotation resulted 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 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 = that x’ = x,y’ = y is obtained, and length OP’ = length OP = r.r.

By paying attention to pervious result, that is tan By paying attention to pervious result, that is tan (180+ (180+ αα) = tan ) = tan αα and and αα (180+ (180+ αα) = cot v then if k is ) = cot v then if k is integer, applies as follows.integer, applies as follows.

sin (k x 360sin (k x 360oo + + αα) = sin ) = sin αα

cos (k x 360cos (k x 360oo + + αα) = cos ) = cos αα

tan (k x 180tan (k x 180oo + + αα) = tan ) = tan αα

sec (k x 360sec (k x 360oo + + αα) = sec ) = sec αα

csc (k x 360csc (k x 360oo + + αα) = csc ) = csc αα

cot (k x 180cot (k x 180oo + + αα) = cot ) = cot αα