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TRIGONOMETRY

TRIGONOMETRY TRIGONOMETRIC RATIOS & IDENTITIES

Gon p sides

Angle (T T)

T–Ratio

1. The meaning of Trigonometry Tri p 3

Trigonometric Ratios of Standard Angles

p

0°

30°

45°

60°

sin

0

1 2

1 2

3 2

1

cos

1

3 2

1 2

1 2

0

tan

0

1 3

1

3

f

cot

f

3

1

1 3

0

sec

1

2 3

2

cosec

f

Metron p Measure

Hence, this particular branch in Mathematics was developed in ancient past to measure 3 sides, 3 angles and 6 elements of a triangle. In today’s time–trigonometric functions are used in entirely different shapes. The 2 basic functions are sine and cosine of an angle in a right–angled triangle and there are 4 other derived functions.

sin T

cos T

tan T

cot T

sec T

cosec T

P H

B H

P B

B P

H B

H P

2

2

2

2 3

90°

f

1

2. Basic Trigonometric Identities (a) sin2T + cos2T = 1 : –1d sinTd 1; –1d cosTd 1  TR (b) sec2T – tan2T = 1 : | secT| t 1  TR (c) cosec2T – cot2T = 1 : | cosecT| t 1  TR

The sign of the trigonometric ratios in different quadrants are as under :

TRIGONOMETRY

3. Trigonometric Ratios of Allied Angles Using trigonometric ratio of allied angles, we could find the trigonometric ratios of angles of any magnitude.

sin (–T) = – sin T

cos (–T) = cos T

§S · sin ¨  T ¸ cos T 2 © ¹

§S · cos ¨  T ¸ sin T 2 © ¹

tan (–T) = – tan T

cot (–T) = –cot T

§S · tan ¨  T ¸ cot T ©2 ¹

§S · cot ¨  T ¸ ©2 ¹

cosec (–T) = – cosec T

sec (–T) = sec T

§S · sec ¨  T ¸ cos ecT 2 © ¹

§S · cos ec ¨  T ¸ sec T 2 © ¹

§S · sin ¨  T ¸ cos T ©2 ¹

§S · cos ¨  T ¸  sin T ©2 ¹

sin S  T

cos S  T

sin T

tan T

 cos T

§S · tan ¨  T ¸  cot T ©2 ¹

§S · cot ¨  T ¸  tan T ©2 ¹

tan S  T

cot S  T

 tan T

 cot T

§S · sec ¨  T ¸  cosecT ©2 ¹

§S · cos ec ¨  T ¸ sec T ©2 ¹

sec S  T

 sec T

cosec S  T

sin S  T

 sin T

cos S  T

cos ec T  cos T

§ 3S · sin ¨  T ¸  cos T 2 © ¹

§ 3S · cos ¨  T ¸  sin T 2 © ¹

tan S  T

cot S  T

tan T

cot T

§ 3S · tan ¨  T ¸ cot T © 2 ¹

§ 3S · cot ¨  T ¸ © 2 ¹

tan T

sec S  T

cosec S  T

 cosec T

 sec T

§ 3S · sec ¨  T ¸  cosec T 2 © ¹

§ 3S · cos ec ¨  T ¸  sec T 2 © ¹

§ 3S · sin ¨  T ¸  cos T © 2 ¹

§ 3S · cos ¨  T ¸ sin T © 2 ¹

sin 2S  T

cos 2S  T

 sin T

cos T

§ 3S · tan ¨  T ¸  cot T 2 © ¹

§ 3S · cot ¨  T ¸  tan T 2 © ¹

tan 2S  T

cot 2S  T

§ 3S · sec ¨  T¸ © 2 ¹

 tan T

cos ec T

 cot T

§ 3S · cos ec ¨  T ¸  sec T © 2 ¹

sec 2S  T

sec T

cosec 2S  T

 cosec T

sin 2S  T

sin T

cos 2S  T

cos T

tan 2S  T

tan T

cot 2S  T

cot T

sec 2S  T

sec T

cosec 2S  T

cosec T

TRIGONOMETRY

4. Trigonometric Functions of Sum or

5. Multiple Angles and Half Angles

Difference of Two Angles (a) sin 2A = 2 sin A cos A ; sin T = 2 sin (a) sin (A + B) = sin A cos B + cos A sin B (b) sin (A – B) = sin A cos B – cos A sin B (c) cos (A + B) = cos A cos B – sin A sin B

(b) cos 2A = cos2A – sin2A = 2 cos2A – 1 = 1 – 2 sin2 A ; 2cos2

T T = 1 + cos T, 2 sin2 = 1 – cos T 2 2

(d) cos (A – B) = cos A cos B + sin A sin B

tan A  tan B 1  tan A tan B

(e) tan (A  B)

(f) tan (A  B)

tan A  tan B 1  tan A tan B

T T cos 2 2

2 tan A ; tan T = (c) tan 2A = 1  tan 2 A

(d) sin 2A =

T 2 2 T 1  tan 2

2 tan

2 tan A 1  tan 2 A ; cos 2A = 2 1  tan A 1  tan 2 A

(e) sin 3A = 3 sin A – 4 sin3 A (f) cos 3 A = 4 cos3 A – 3 cos A

(g) cot (A + B) =

cot A cot B  1 cot B  cot A

cot A cot B  1 (f) cot (A - B) = cot B  cot A

(h) sin2 A – sin2 B = cos2B – cos2A = sin (A + B) . sin (A – B) (i) cos2 A – sin2 B = cos2B – sin2A = cos (A + B) . cos (A – B)

(j) tan (A + B + C) = tanA  tanB  tanC  tanAtanBtanC 1  tanAtanB  tanBtanC  tanCtanA

(g) tan 3A =

3tan A  tan  A 1  3tan 2 A

6. Transformation of Products into Sum or Difference of Sines & Cosines

(a) 2 sin A cos B = sin (A + B) + sin (A – B) (b) 2 cos A sin B = sin (A + B) – sin (A – B) (c) 2 cos A cos B = cos (A + B) + cos (A – B) (d) 2 sin A sin B = cos (A – B) – cos (A + B)

TRIGONOMETRY

7. Factorisation of the Sum or Difference of

9. Conditional Identities

Two Sines or Cosines

If A + B + C = S then : (i) sin 2A + sin2 B + sin 2C = 4 sin A sin B sin C

(a) sin C + sin D = 2 sin

C D CD cos 2 2

(ii) sin A + sin B + sin C = 4 cos

A B C cos cos 2 2 2

(iii) cos 2A + cos 2B + cos 2C = –1 – 4cosA cosB cosC

CD CD (b) sin C – sin D = 2 cos sin 2 2

(iv) cos A + cos B + cos C = 1 + 4sin

(v) tan A + tan B + tan C = tanA tanB tanC

CD CD (c) cos C + cos D = 2 cos cos 2 2

(d) cos C – cos D = – 2 sin

(vi) tan

CD CD sin 2 2

A B B C C A tan  tan tan  tan tan 1 2 2 2 2 2 2

(vii) cot

A B C  cot  cot 2 2 2

cot

A B C .cot .cot 2 2 2

(viii) cot A cot B + cot B cot C + cot C cot A = 1

8. Important Trigonometric Ratios (a) sin n S = 0 ; cos n S = (–1)n ; tan nS = 0 where n Z

A B C sin sin 2 2 2

10. Range of Trigonometric Expression E = a sin T + b cos T

(b) sin 15º or sin

S 12

3 1 5S ; = cos 75º or cos 12 2 2

3 1 S 5S ; cos 15º or cos = = sin 75º or sin 12 12 2 2

E

§ a   b 2 sin(T  D), ¨ where tan D ©

b· ¸ a¹

E

§ a 2  b 2 cos(T  E), ¨ where tan E ©

a· ¸ b¹

Hence for any real value of T,  a 2  b 2 d E d a 2  b 2 tan 15º =

tan 75º =

(c) sin

3 1 3 1

3 1 3 1

2  3 = cot 75º ;

11. Sine and Cosine Series (a)

nE 2 sin (D  n  1 E) = E 2 sin 2 sin

2  3 = cot 15º

S or sin 18º = 10

5 1 & 4

sin D + sin (D + E) + sin (D + E) + ..... + sin (D + n  1 E )

(b)

cos D + cos ( D + E) + cos (D + 2E) + ...... + cos (D + n  1 E ) nE 2 cos (D  n  1 E) = E 2 sin 2 sin

cos 36º or cos

S 5

5 1 4

TRIGONOMETRY

12. Graphs of Trigonometric Functions (a)

(d)

y = cot x, x  R – {nS; n  z}; y  R

(e)

y = cosec x, x R – {nS; n Z}; y  –f–]‰[1, f)

y = sin x, x  R ; y  [–1, 1]

(b)

y = cos x, x  R ; y  [–1, 1]

(f)

(c)

y = tan x,

S ­ ½ x  R  ® 2n  1 ; n  Z ¾ ; y  R 2 ¯

y = sec x,

S ­ ½ x  R  ® 2n  1 ; n  Z ¾ ; y  –f–]‰[1, f) 2 ¯

TRIGONOMETRY

TRIGONOMETRIC EQUATIONS

4.

ª S Sº D  « , » ¬ 2 2¼

13. Trigonometric Equations The equations involving trigonometric functions of unknown angles are known as Trigonometric equations. e.g.,

cos T= 0, cos2T– 4 cos T= 1.

5.

cos T= cos DœT= 2nS± D, where D[0, S].

6.

§ S S· tan T= tan DœT= n S+ D, where D  ¨  , ¸ © 2 2¹

7.

sin2T= sin2 DœT= n S± D.

8.

cos2 T= cos2 DœT= n S± D.

9.

tan2 T= tan2 DœT= n S± D.

A solution of a trigonometric equation is the value of the unknown angle that satisfies the equation.

e.g.,

sin T

1 ŸT 2

S or T 4

sin T = sin D œ T = n S + ( – 1) n D, where

S 3S 9S 11S , , , ,... 4 4 4 4

Thus, the trigonometric equation may have infinite number of solutions and can be classified as :

10. sin T= 1 œT= (4n +1)

S . 2

(i)

Principal solution

11. cos T= 1 œT= 2n S.

(ii)

General solution

12. cos T= – 1 œT= (2n + 1) S.

14. General Solution

13. sin T= sin Dand cos T= cos DœT= 2n S+ D.

Since, trigonometric functions are periodic, a solution generalised by means of periodicity of the trigonometrical functions. The solution consisting of all possible solutions of a trigonometric equation is called its general solution.

14.1 Results

1. Every where in this chapter ‘n’ is taken as an integer, if not stated otherwise.

1.

sin T= 0 œT= n S

2.

S cos T= 0 œT(2n + 1) 2

3.

tan T= 0 œT= n S

2. The general solution should be given unless the solution is required in a specified interval or range. 3. D is taken as the principal value of the angle. (i.e., Numerically least angle is called the principal value).

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