Chapter - 33

Trigonometric Ratios and Identities

1. AngleConsider a ray OA. If this ray rotates about its end-point O andtakes the position OB, then we say that the angle AOB has beengenerated.

Thus, an angle is considered as the figure obtained by rotating agiven ray about its end-point.The revolving ray is called the generating line of the angle. Theinitial position OA is called the initial side and the final positionOB is called the terminal side of the angle. The end-point Oabout which the ray rotates is called the vertex of angle.

2. Positive and Negative AnglesIf the ray OA rotates in anticlockwise direction a positive angleis formed and when the ray OA rotates in clockwise directionnegative angle is formed.

3. Four QuadrantsLet XOX and YOY be two lines perpendicular to each other. Theintersecting point O is called the origin. The lines XOX andYOY are respectively called x-axis and y-axis. These lines dividethe plane into four parts called the quadrants. The parts XOY,YOX, XOY and YOX are known as first, second, third and fourthquadrants respectively as shown in the figure. An angle is saidto be in a particular quadrant, if the terminal side of the anglelies in that quadrant.

224 Magical Book on Arithmetical Formulae

4. Different Units For Measuring AnglesThere are three systems for measuring angles in trigonometry.(i) Sexagesimal or English System: In this system a right

angle is sub-divided as under,(a) One right angle = 90 degrees, symbolically written as 90.(b) One degree = 1 = 60 minutes, symbolically written as

60.(c) One minute = 1 = 60 seconds, symbolically written as

60.The unit of measurement in this system is degree.

(ii) Centesimal System or the French System: In this system,we have,(a) One right angle = 100 grades, written as 100g.(b) One grade = 1g = 100 minutes, written as 100.(c) One minute = 1 = 100 seconds, written as 100.The unit of measurement in this system is gradian or grades.Note: Formula for conversion from English to French system

is 90 = 100g(iii)Circular System: In this system angle is measured in

radians and we have,

1 right angle = 2

radians

It is also called circular measure or radian measure.Note:(a) The angle subtended at the centre of a circle by an arc of

length equal to its radius is 1 radian, written as 1c.c = 180 = 2 right angles

1c = 180

and 1 = c

180

(b) Value of = 227 or = 3.1416 nearly.

(c) The units of measurement in the circular system isradian.

Trigonometric Ratios and Identities 225(d) Radian measure of some common angles are:

Angles in Degrees 30 45 60 90 180 270 360

Angles in Radians 6

4

3

2

23

2

(e) 1 radian = 57 1622 (approximately)(f) 1 degree = 0.01746 radian

5. Relation Between Angle, Radius and Arc LengthIf (radian) is the angle made by an arc of length s at the centre

of a circle of radius r, then = rs

Note: While using the above result = rs

, the angle must be

expressed in radians, if given in any other unit.

6. Trigonometric RatiosThe most important task of trigonometry is to find the remainingsides and angles of a triangle when some of its sides and anglesare given. This problem is solved by using some ratios of thesides of a triangle with respect to its acute angles. These ratiosof acute angles are called trigonometric ratios of angles.

Consider an acute angle YAX = with initial side AX and terminalside AY. Let P be any point on the terminal side AY. PMperpendicular from P on AX to get the right-angled triangle AMPin which PAM = .In the right-angled triangle AMP, Base = AM = b, Perpendicular =PM = p and Hypotenuse = AP = h.We define the following six trigonometric ratios:

(i) Sine = PerpendicularHypotenuse

ph

, and is written as sin.

226 Magical Book on Arithmetical Formulae

(ii) Cosine = Base

Hypotenusebh

, and is written as cos.

(iii) Tangent = Perpendicular

Basepb

, and is written as tan.

(iv) Cosecant = Hypotenuse

Perpendicularhp

, and is written as cosec.

(v) Secant = Hypotenuse

Basehb

, and is written as sec.

(vi) Cotangent = Base

Perpendicularbp

, and is written as cot.

Note:(a) It should be noted that sin is an abbreviation for sine

of angle , it is not the product of sin and .Similar is the case for other trigonometric ratios.

(b) The above trigonometric ratios are defined for an acuteangle .

(c) The trigonometric ratios are same for the same angle.

7. Relations Between Trigonometric RatiosThe trigonometric ratios sin, cos and tan of an acute angle are very closely connected by a relation. If any one of them isknown, the other two can be easily calculated. Now, look at thesome important formulae given below:

(i) (a) sin = 1

cosec (ii) (a) cos = 1

sec

(b) cosec = 1

sin (b) sec = 1

cos (c) sin. cosec = 1 (c) cos. sec = 1

(iii) (a) tan = 1

cot (iv) (a) tan = sincos

(b) cot = 1

tan (b) sin = tan . cos

(c) tan. cot = 1 (c) cos = sintan

(v) (a) cot = cossin

(b) cos = cot . sin

(c) sin = coscot

Trigonometric Ratios and Identities 2278. Trigonometric Ratios of Some Special Angles

Values of the trigonometric ratios for some special angles aregiven below:

9. Trigonometric Ratios of Complementary AnglesWe know that two angles are said to be complementary, if theirsum is 90. Thus, and (90 ) are complementary angles for anacute angle .If is an acute angle, then(i) sin (90 ) = cos (ii) cos (90 ) = sin (iii) tan (90 ) = cot (iv) cot (90 ) = tan (v) sec (90 ) = cosec (vi) cosec (90 ) = sec

10. Trigonometric IdentitiesWe know that an equation is called an identity if it is true for allvalues of the variable(s) involved. For example, x2 4 = (x 2)(x + 2) is an algebraic identity as it is satisfied by every value ofthe variable x.Similarly, an equation involving trigonometric ratios of an angle(say) is said to be a trigonometric identity if it is satisfied for allvalues of for which the given trigonometric ratios are defined.

For example, sin2 12 sin = sin

1sin2

is a trigonometric

228 Magical Book on Arithmetical Formulae

identity, whereas sin 1sin2

= 0 is an equation. Also,

sec = 1

cos is a trigonometric identity, because it holds for all

values of except for which cos = 0. For cos = 0, sec is notdefined.Following are some fundamental trigonometric identities:(i) (a) sin2 cos2 = 1

(b) sin2 = 1 cos2 (c) cos2 = 1 sin2

(ii) (a) sec2 = 1 + tan2 (b) sec2 tan2 = 1(c) sec2 1 = tan2

(d) sec+ tan = 1

sec tan

(e) sec tan = 1

sec tan (iii) (a) cosec2 = 1 + cot2

(b) cosec2 cot2= 1(c) cosec2 1 = cot2

(d) cosec+ cot = 1

cosec cot

(e) cosec cot = 1

cosec cot 11. Trigonometric Ratios in Terms of other Trigonometric

Ratios

Trigonometric Ratios and Identities 22912. Sign of Trigonometric Ratios

(i) First Quadrant: All trigonometric ratios are positive.(ii) Second Quadrant: sin and cosec are positive.(iii) Third Quadrant: tan and cot are positive.(iv) Fourth Quadrant: cos and sec are positive.

Remember: I II III IVAll sin tan cos

13. Range of Trigonometric Ratios(i) 1 < sin < 1 and 1 < cos < 1. Thus |sin| < 1 and |cos|

< 1.The value of sin is never greater than 1 and never lessthan 1. The value of cos is never greater than 1 and neverless than 1.

(ii) cosec > 1 and cosec < 1.(iii) sec > 1 and sec < 1.(iv) tan can assume any value.

14. Trends of Trigonometric Ratios in Various QuadrantI. First Quadrant (As increases from 0 to 90)

sine : increases from 0 to 1cosine : decreases from 1 to 0tangent : increases from to cotangent : decreases from to 0secant : increases from 1 to cosecant : decreases from to 1

II. Second Quadrant (As increases from 90 to 180)sine : decreases from 1 to 0cosine : decreases from 0 to 1tangent : increases from to 0

230 Magical Book on Arithmetical Formulaecotangent : decreases from 0 to secant : increases from to 1cosecant : increases from 1 to

III. Third Quadrant (As increases from 180 to 270)sine : decreases from 0 to 1cosine : increases from 1 to 0tangent : increases from 0 to cotangent : decreases form to 0secant : decreases from 1 to cosecant : increases from to 1

IV. Fourth Quadrant (As increases from 270 to 360)sine : increases from 1 to 0cosine : increases from 0 to 1tangent : increases from to 0cotangent : decreases form 0 to secant : decreases from to 1cosecant : increases from 1 to

15. Trigonometric Ratios of Negative, Complementary andSupplementary AnglesThe following results are useful for finding the values oftrigonometric ratios in various quadrants.(i) sin() = sin, tan() = tan, sec() = sec

cos() =cos, cot() = cot ,cosec() = cosec

(ii) (90 ) lies in Ist Quadrant. Here, all the t-ratios arepositive. Therefore,sin (90 ) = cos, tan(90 ) = cot,sec (90 ) = coseccos (90 ) = sin, cot (90 ) = tan,cosec (90 ) = sec

(iii) (90 + ) lies in IInd Quadrant. Here, sin and cosec arepositive and rest are negative. Therefore,sin (90 + ) = cos, tan(90 + ) = cot,sec(90 +) = cosec cos (90 + ) = sin, cot (90 + ) = tan,cosec(90 + ) = sec

(iv) (270 ) lies in IIIrd Quadrant. Here, tan and cot are positiveand rest are negative. Therefore,sin (270 ) = cos, tan (270 ) = cot,sec (270 ) = coseccos (270 ) = sin, cot (270 ) = tan,cosec (270 ) = sec

(v) (270 + ) lies in IVth Quadrant. Here, cos and sec arepositive and rest are negative. Therefore,

Trigonometric Ratios and Identities 231sin (270 + ) = cos, tan (270 + ) = cot,sec (270 + ) = coseccos (270 + ) = sin, cot (270 + ) = tan,cosec (270 + ) = sec

(vi) (180 ) lies in IInd Quadrant. Here, sin and cosec arepositive and rest are negative. Therefore,sin (180 ) = sin , tan (180 )= tan, sec (180 ) = seccos (180 ) = cos , cot (180 ) = cot ,cosec (180 ) = cosec

(vii) (180 + ) lies in IIIrd Quadrant. Here, tan and cot arepositive and rest are negative. Therefore,sin (180 + )= sin, tan (180 + )= tan, sec (180 + ) = sec cos (180 + ) = cos , cot(180 + )= cot , cosec (180 + ) = cosec

(viii) (360 ) lies in IVth Quadrant. Here, cos and sec arepositive and rest are negative. Therefore,sin (360 ) = sin, tan(360 ) = tan ,sec (360 ) = sec cos(360 ) = cos , cot(360 ) = cot,cosec(360 ) = cosec Note:It may be noted that final position of revolving line forthe angle (360 ) occupy the same position as forsame angle (). Therefore, t-ratios of (360 ) have thevalue as of ().

(ix) (360 + ) lies in Ist Quadrant. Here, all the t-ratios arepositive. Therefore,sin (360 + ) = sin , tan (360 + ) = tan ,sec (360 + ) = sec cos (360 + ) = cos, cot (360 + ) = cot,cosec (360 + ) = cosec

16. Sum and Difference Formulae For Trigonometric Ratios(i) sin (A + B) = sinA cosB + cosA sinB(ii) sin (A B) = sinA cosB cosA sinB(iii) cos (A + B) = cosA cosB sinA sinB(iv) cos (A B) = cosA cosB + sinA sinB(v) sin (A + B) + sin (A B) = 2 sinA cosB(vi) sin (A + B) sin (A B) = 2 cosA sinB(vii) cos (A + B) + cos (A B) = 2 cosA cosB(viii) cos (A B) cos (A + B) = 2 sinA sinB(ix) sin2A sin2B = sin (A + B) sin (A B)(x) cos2A sin2B = cos (A + B) cos (A B)

232 Magical Book on Arithmetical Formulae17. Some More Formulae

(i) sinA + sinB = 2 sinA B

2

cosA B

2

(ii) sinA sinB = 2 cosA B

2

sinA B

2

(iii) cosA + cosB = 2 cosA B

2

cosA B

2

(iv) cosA cosB = 2 sinA B

2

sinA B

2

18. Tangent Formulae

(i) tan (A + B) = tan A tanB1 tan.tanB

(ii) tan (A B) = tan A tanB

1 tanA tanB

19. Trigonometric Ratios of Multiple Angles(i) sin2A = 2 sinA cosA(ii) cos2A = cos2A sin2A = 2 cos2A 1 = 1 2 sin2A(iii) 1 + cos2A = 2 cos2A and 1 cos2A = 2 sin2A(iv) sin3A = 3 sinA 4 sin3A(v) cos3A = 4 cos3A 3 cosA

(vi) tan2A = 22tan A

1 tan A

(vii) tan3A = 3

2

3tanA tan A1 3tan A

(viii) cos2A = 2

2

1 tan A1 tan A

(ix) sin2A = 22tan A

1 tan A

20. Trigonometric Ratios of Sub-multiple Angles

(i) sinA = 2 sinA2 cos

A2

(ii) cosA = cos2A2 sin

2A2 = 2 cos

2A2 1 = 1 2sin

2A2

Trigonometric Ratios and Identities 233

(iii) tanA = 2

A2tan2A1 tan2

(iv) 1 cosA = 2 sin2A2 and 1 + cosA = 2cos

2A2

(v) sinA = 2

A2tan2A1 tan2

(vi) cosA =

2

2

A1 tan2A1 tan2

21. Trigonometric Ratios of Some Special Angles

(i) sin15 = 3 12 2

= cos75

(ii) cos15 = 3 12 2

= sin75

(iii) sin18 = 5 14

= cos72

(iv) cos18 = 10 2 54 = sin72

(v) sin36 = 10 2 54 = cos54

(vi) cos36 = 5 14

= sin54

(vii) sin1222

= 2 2

2

(viii) cos1222

= 2 2

2

22. Trigonometric Equations(i) sin = 0 = n

(ii) cos = 0 = (2n+ 1) 2

(iii) tan = 0 = n

234 Magical Book on Arithmetical Formulae23. Periodicity

A function f(x) has periodicity p, if f(x + p) = f(x).sin x has periodicity 2, since sin (x + 2) = sin xcos x has periodicity 2, since cos (x + 2) = cos x

24. Important Points to Remember(i) The values of the trigonometric ratios of an angle do not

vary with the length of the sides of the triangle, if angleremains the same.

(ii) Maximum value of m sin n cos = 2 2m nFor example, maximum value of

3 sin + 4 cos = 2 23 4 = 16 9 = 25 = 5

(iii) Maximum value of m sin n sin = 2 2m n

(iv) Maximum value of m cos n cos = 2 2m n

(v) Minimum value of m sin n cos = 2 2m nFor example, minimum value of

sin cos = 2 21 1 = 2(vi) (a) sin1.sin2.sin3.sin4..............sin180 = 0

(b) sin1 . sin2 . sin3.sin4.............. to(greater than sin180) = 0

(vii) (a) cos1.cos2.............cos90 = 0(b) cos1.cos2.............to (greater than cos90) = 0

(viii) (a) tan1.tan2.............tan89 = 1(b) cot1.cot2..............cot89 = 1

(ix) If sec tan = x, then sec = 2 12

xx

(x) If sin + cos = x, then sin cos = 22 x(xi) If sin + cosec = x, then sinn + cosecn = x(xii) If tan + cot = x, then tann + cotn = x(xiii) It should be noted that sin2 = (sin)2, sin3 = (sin)3, cos3

= (cos)3, etc.