EXERCISE- A
Find the value of:
1) sin2760. - √3/2
2) sin 1755 - 1/√2
3) sin(-3330). -1
4) sin120. √3/2
5) sin 4530. - 1/2
6) Sin 150°. 1/2
7) Sin(-11π/4). -1/√2
8) cosec (-1410). 2
9) cosec 675. -√2
10) Cosec(-675). √2
11) Cosec(-660). 2/√3
12) Cosec(16π/3). -2/√3
13) cos(-1170). 0
14) cos 315. 1/√2
15) cos 855. -1/√2
16) cos 690. √3/2
17) Cos(5π/2- 19π/3). √3/2
18) sec 210. -2/√3
19) Sec(-31π/4). √2
20) sec 11π/4. -√2
21) Sec(15π/4). √2
22) Sec 15π/4. √2
23) tan(-1755). 1
24) Tan(-1125). -1
25) tan(-17π/4). -1
26) Tan(3π/2+ π/3). -1/√3
27) cot 1230. -√3
28) cot(-870). √3
29) cot 330. -√3
30) Cot 840. -1/√3
31) Cot(16π/3). 1/√3
32) cot(-1575). 1
EXERCISE - B
Express in terms of TRIGONOMETRICAL ratio of a positive angles less than 45°:
1) sin 139°29'30". sin 40°39'30"
2) sin(-7495). Cos 25
3) sin 194. - sin14
4) sin 348. - sin 12
5) sin (-1785). sin 15
6) cosec(-830). -sec 20
7) cosec(-1324). Sec 26
8) cosec(-7498). Sec 28
9) cos 189. - cos 9
10) sec(-227'5°). - cosec 42'5°
11) sec(-1875). Cosec 15
12) tan 615. cot 15
13) tan 305. -cot 35
14) tan 3598. -tan 2
15) cot(-1952). Cot 28
16) cot(-1358). Tan8
EXERCISE - C
Express in terms of the Trigonometrical ratios of x:
1) sin(x- 450). - cosx
2) sin(π+x). - sinx
3) cosec(x-3π/2). Secx
4) cos(x - 450). Sinx
5) tan(-π/2 - x). Cotx
6) cot(540 - x). -cot x
EXERCISE - D
Find the values of:
1) sin135 cos315+ sin420 cos320. √3/2
2) √3sin(1380)+tan²(-240)- cos²(405). 1
3) sin 330 + tan 45 - 4 sin²120+ 2 cos² 135 + sec²180. -1/2
4) sin480 cos690 +cos780 sin1050. 1/2
5) sin²120 + cos²150 + tan²120 + cos 180 - tan 135. 9/2
6) sin²(-300)cos³(120) + cos²(-240) sin³(390). -1/16
7) sin750 cos300 + cos1470 sin (-1020). 1
8) {sin150 - 5cos 300+ 7tan 225}/(tan 135 +3 sin 210). -2
9) {sin(90-x)sec(180-x)sin(-x)}/ {sin(180+x) cot(360-x)cosec(90+x)}. Sinx
10) m² sin π/2 - n² sin 3π/2 + 2mn sec π. (m-n)²
11) {sin(270+A) cos(90- A)}/{sin(180-A) cos(180- A)}. 1
12) cos570 sin510+ sin(-330)cos(-390). 0
13) cos 24+ sin 55 + cos125 + cos 204 + cos(300). 1/2
14) {cos(90+x) sec(-x) tan(180-x)}/{sec(360+x)sin(180+x)cot(90-x)}. -1
15) cosx/{sin(90+x)} + sin(-x)/{sin(180+x)} - {tan(90-x)}/cotx. 3
16) cos(-x)/{sin(90+x)}. 1
17) (cos 3x - 2 cos 4x)/(sin 3x + 2sin 4x), when x= 150. -1/2(1+√3)
18) {sec(270+x)sec(90-x)+ tan(270-x) tan(90+x)}/{cotx + tan(180+x) + tan(90+x)+ tan(360-x) + cos180}. -1
19) tan π/4+ tan 3π/4 + tan 5π/4 + tan 7π/4. 0
20) tan(-x)/{sin 540)}. secx
EXERCISE - E
PROVE:
1) sin780 sin480+ sin30 cos120= 1/2.
2) sin780 cos 390- sin330 cos(-300)= 1.
3) sin420cos390+ cos(-390)sin(-330)=1.
4) (sin250 + tan 290)/(cot 200+ cos340)= 1.
5) 3[sin⁴(3π/2 - x) +sin⁴(3π+x)] - 2 [sin⁶(π/2 +x)+ sin⁶(5π-x)]= 1.
6) sin420 cos390+ cos(-300)sin(-330)=1.
7) (sin150 - 5cos300 +7 tan225)/(tan 135 + 3 sin 210) = -2.
8) sin²54 - sin²72 = sin² 18 - sin²36.
9) Sin(π/2 + x) cos(π- x) cot(3π/2 + x) = sin(π/2- x) sin(3π/2 - x) cot(π/2+ x).
10) cos 306+ cos234+ cos162+cos18= 0.
11) {cos²(π/2 +x)}/[{sec²(π+x)} - 1] + {cos²(2π -x)}/{cosec²(π+x)}- 1]= 1.
12) 4 cos²210 - tan 315 + 4 cosec 90 = 8.
13) cos 24 + cos 55 + cos 125 + cos 204 + cos 270 + cos 300= 1/2
14) tan 225 cot 405+ tan765 cot 675 = 0.
15) tan130 tan 140 = 1.
16) {1+ cotx - sec(π/2+ x)}{1+ cotx + sec(π/2+ x)}= 2 cotx.
17) cot(9/-x) cotx cos(90-x) tan(90- x) = cosx.
EXERCISE - F
Show that following are are independent of x:
1) {sin(π/2 -x) sin(3π/2 - x) cot(π/2+ x)}/(sinx cos x).
2) sin(π/2+ x) sin(π+ x)+ cos(π/2+ x) cos(π- x).
3) {sin³(2π - x)}/{cos²(3π/2 +x)}. {Cos³(2π- x)}/{sin³(2π+x)} . {tan(π-x)}/{cosec²(π-x)}.{sec²(π+x)}/sinx.
4) 3{sin⁴(3π/2 - x) + sin⁴(3π + x)} - 2{sin⁶(π/2+ x) + sin⁶(5π- x)}.
5) cotx + tan(180+x)+ tan(90+x) + tan(360-x).
EXERCISE - G
1) In any triangle ABC, Prove that:
a) sin A cos (B+C)+ cosA sin(B+C)= 0.
b) sin(B+ C)+ sin(C+ A)+ sin(A+ B)= sinA + sinB + sinC.
c) {sin(B+ C)+ sin(C+ A)+ sin(A+ B)}/{sin(π+A) + sin(3π+B) + sin(5π+C)= -1
d) Sin(A+ B) - cosC = cos(A+ B)+ sinC.
e) cos(A+B)/2= sin C/2.
f) cos C = - cos(A+B).
g) Cos A cos C + cos(A+ B) cos(B+C)} /{cosA sin C - sin(A+B) cos(B+C) = cot C.
h) tan A/2= cot{180 + (B+C)/2}.
i) {tan (B+C)+tan (C+A)+ tan(A+B)} /{tan(π+A)+ tan(3π+B)+ tan(5π+C) = -1
j) tan (B+C)/2 = cot A/2.
k) tan (B+C)+tan (C+A)+ tan(A+B)}/ {tan(π-A)+ tan(2π-B)+ tan(3π-C) = 1.
l) cot A tan (B+C)- cosA sec(B+C)= 0
2) If A, B, C, D are the angles of a quadrilateral, prove:
1) cos(A+B)/2 + cos(C+D)/2= 0.
2) cos (A+B)/2 + cos(C+ D)/2= 0.
3) cos ((A+C)/2 + cos(B+ D)/2= 0.
4) tan (A+ B)/2 + tan (C+D)/2= 0.
3) If A, B, C, D are the four angles, taken in this order, of a cyclic quadrilateral, prove that:
1) cos A + cosB + cosC + cosD= 0.
2) tan(A+B)+ tan(C+ D)= 0
3) cot A + cotB + cotC + cotD= 0
EXERCISE - H
If n is any integer, find the value of:
1) sin{nπ +(-1)ⁿ π/3}. √3/2
2) {1+ sin (π- x) cos nπ}{1+ sin (π+x) cos nπ} cos² x
3) Sin(-1230) - cos{(2n +1)π +π/3}. 0
4) cosec{nπ +(-1)ⁿ π/4}. ± √2
5) cosec{nπ/2 +(-1)ⁿ π/6}. ±2, ±2/√3
6) cos nπ. (-1)ⁿ
7) cos{nπ +(-1)ⁿ π/3}. ±1/2
8) Cos(nπ+ x). (-1)ⁿ cosx
9) tan{nπ +(-1)ⁿ π/4}. 1
10) tan{nπ +x}. (-1)ⁿ tanx.
11) Tan(nπ - x). - tanx
EXERCISE - I
1) If tan 25= a, then show that (tan 205 + tan 295)/(tan 65- tan 335) = (a² -1)/(a² +1).
2) If tan 25= x, then show that (tan 155 - tan 115)/(1+tan 155. tan 115) = (1-x²)/2x.
EXERCISE- J
Prove:
1) sin²π/8 + sin² 3π/8+ sin² 5π/8 + sin² 7π/8 =2.
2) sin²π/4 + sin² 3π/4 + sin² 5π/4 + sin² 7π/4 =2.
3) cos²π/4 + cos²3π/4+ cos²5π/4 + cos²7π/4 = 2.
4) cos²π/8 + cos²3π/8+ cos²5π/8 + cos²7π/8 = 2.
5) tan π/16 tan 3π/16 tan 5π/16 tan 7π/16 = 1.
6) tan π/12 tan 5π/12tan 7π/12 tan 11π/12 = 1.
7) cot π/20 cot 3π/20 cot 5π/20 cot 7π/20 cot 9π/20 = 1
EXERCISE - K
Solve:
1) 2 Sinx -1= 0. 30
2) 2 sin 3x =1. 10
3) 3 Sinx = 2 cos²x. 30
4) 2 sin 2x = 1/√3. 30
5) 2 sin 2x = √3. 30
6) 2 sin²x =1/2. 30
7) Sin²x - 2 cosx + 1/4 = 0. 60
8) 4 sin²x - 3 = 0. 60
9) Sin²x - 1/2 Sinx = 0. 0 , 30
10) Sinx/(1- cosx) + Sinx/(1+ cosx)= 4. 30
11) Sinx= coax. 1
12) 2 sin²x = 1/2. 30
13) 2 cosx = 1. 60
14) 2 cos 3x = 1. 20
15) Cosx/(1- Sinx) + cosx/(1+ Sinx)= 4. 60
16) Cosx/(cosecx +1) + cosx/(cosec -1)= 2. 45
17) 2 cos²x = 1/2. 60
18) 2 cos²x -1= 0. 45
19) 2 cos²x + Sinx - 2=0. 0 , 30
20) Cos²x - 1/2 cosx = 0. 90,60
21) (Cos²x - 3 cosx +2)/sin²x = 2. 90
22) Cos²x/(cot²x - cos²x) = 3. 60
23) Tan 5x = 1. 9
24) 3 tanx + cotx = 5 cosecx 60.
25) 3 tan²x -1= 0. 30
26) 4 tan²x =12. 30
27) Tan²x + 3 = 3 Secx. 0,60
28) 3 tan²2x -1= 0. 15
29) Tan²x + 3= 3 Secx. 0,60
30) Tan²x + cot²x = 2. 45
31) Tan²x - (√3 +1)+ √3 = 0. 45, 60
32) Sec²x - 2 Tanx = 0. 45
33) Sec²x + tan²x = 5/3. 30
34) Cosec²x - cotx (1+√3)+ √3 -1= 0. 45, 60
Mg. A- R.1
1) Find the value of:
a) cos(-1170°). 0
b) cos(-870°). √3
c) tan(-1755°). 1
d) cot 660° + tan(-1050°). 0
e) sec(-945°). -√2
f) cosec(-840°). -2/√3
g) sin 135° cos 210° tan 240° cot 300° sec 330°. 1/√2
h) cos 24°+ cos 55° + cos 125° + cos 204° + cos 300°. 1/2
Mg. A- R.2
Find the value of:
1) tan π/12 tan 5π/12 tan 7π/12 tan 11π/12. 1
2) tan 1° tan 2° tan 3°....... tan 87° tan 88° tan 89°. 1
3) sin²120+ cos²120+ tan²120+ cos180 - tan 135. 9/2
4) If A, B, C, D are the successive angles of a cyclic quadrilateral then prove
i)cosA+ cos B + cos C + cos D = 0.
ii) tan(A+ B)+ tan(C + D)= 0.
5) If cos x - sin x =√2 sin x then show cosx + sinx =√2 cosx.
6) If x= r cos k cos m, y= r cos k sin m and z= r sin k then show x² + y² + z² = r².
7) If tan⁴x + tan²x = 1, then show cos⁴x + cos²x = 1.
8) If x= a sec k cos m, y= b sec k sin m and z= c tan k, then show x²/a² + y²/b² - z²/c² = 1.
9) if tan x= (siny - cos y)/(sin y + cos y) then show sin y + cos y= ± √2 cos k.
Mg. A- R.3
1) If x be an angle of fourth quadrant and sec x= 5/3 then find the value of (6tan x + 5 cosx)/(5 cotx + cosecx). 1
2) If tanA + sin A= m and tanA- sin A= n, then show m² - n² = 4√(mn).
3) If 3 sinx + 4 cosx = 5, then show sin x= 3/5.
4) If tan x + sinx = m, tan x - sinx = n, then show mn= tan²x. sin²x and 4√(mn)= m² - n².
5) show 3[sin⁴(3π/2 - x) + sin⁴(3π+ x)] - 2[sin⁶(π/2 + x) + sin⁶(5π - x)] independent of x.
6) If sinx + cosecx = 2, then show sin⁷x + cosec⁷x = 2.
7) If 1 + sin²A = 3 sinA cosA then find the value of tan A. 1, 1/2
8) A, B, C are the three angles of an acute angled triangle and cos(B+ C - A)= 0, sin(C+ A- B)= √3/2. Find the value of A, B, C. 45, 60, 75
Mg. A- R.4
1) Find the value of m² sin(π/2) - n² sin(3π/2) + 2mn sec π. (m - n)²
2) tan(π/4)+ tan(3π/4)+ tan(5π/4)+ tan(7π/4). 0
3) If 6x= 11π, the value of 2 cosx + 3 tan x is
a) 1 b) 0 c) √3 d) 2√3
4) If cot x= cos 60+ sin 30, The value of cosx + cos(x -90) is
A) √2 B) 1/√2 c) 1 d) 0
5) If 3 sin²x + 5 cos²x= 4 and π/2< x < π, find the value of sin 2x. -1
6) If 3 tanx = -4/3 , find the value of sin x. -4/5 or 4/5
7) If sinx = -2/3 and 270°< x < 360°, find the value of sin (x-270°) tan(360° - x). 2/3
Mg. A- R.5
1) If A, B, C, D are the four angles of a Quadrilateral, show that Sin(A+ B)+ sin(C+ D)= 0
2) A, B, C are the angles of a triangle, show that tan (C- A)/2 = cot (A + B/2)
3) If A+ B =60, show that, sin(120- A)= cos(30- B).
4) Find the value of tan(nπ +π/4). 1
5) If x=100°, determine the sign of the expression Sinx + cosx. Positive
6) If sin(A- B)=√3/2 and sinA=1/√2, find the positive value of B. 75°
7) If cosx= cos y (x≠ y), find the possible value of cos(x+ y). 1
8) Prove that, tan 1° + tan 2°.... Tan 88° tan 89° = 1
9)Find the value of cos 1 + cos 2 + cos 3+ .....+ Cos 180. -1
Mg. A- R.6
1) Prove that sec(-1680). Sin 330=-1.
2) If A, B, C, D are the four angles of a cyclic Quadrilateral, show that cosA+ cos B + cos C+ coa D= 0.
3) If tan 25= a, prove that (tan 155 - tan 115)/(1+ tan 155. Tan 115)= (1- a²)/2a
4) If A, B, C are the angles of a triangle, show that {Sin(B+ C)+ sin(C+ A) + sin(A+ B)}/{Sin(π+A)+ sin(3π+B)+ sin(5π+C)}= -1
5) If Secx =√2 and 3π/2< x < 2π, find the value of (1+ tanx + cosecx)/(1+ cotx - cosecx). 1
6) If tan x= 0.4, when x lies between 0° and 360°, write down the possible values of x and Sinx. 21°47', 201°48', 0.3714, -0.3714
l) If cos²x - sin²x = tan²y, then show cos²y - sin²y = tan²x.
m) Show: 4(sin⁶x + cos⁶x) - 6(sin⁴x + cos⁴x)= - 2.
n) If x sin³a + y cos³a= sina and x sina - y cosa = 0, then prove x² + y² = 1.
o) If a sinx = b cos x = (2c tanx)/(1- tan²x), prove (a² - b²)²= 4c²(a² + b²).
p) If a sinx + b cos x = c then show a cosx + b sinx = ±√(a²+ b²+c²)
q) If 4x secA = 1+ 4x² then show secA + tanA = 2x or 1/2x.
r) If cos⁴x + cos²x = 1 then prove tan⁴x + tan²x = 1.
s) If p tanx = tan px then show sin²px/sin²x = p²/{1+ (p² -1)sin²x}
t) Eliminate x: tanx - cotx = a, cosx + sinx = b.
u) If tanA= n tan B, sinA= m sin B, then show cos²A = (m² -1)/(n² -1).
IDENTITY AND ASSOCIATED ANGLE
Miscellaneous Chapter-2
1) Eliminate x:
A) a= c(secx + tanx) and b= c(secx - tanx)
B) a sinx + b cosx = c and a cosx - b sinx = d.
C) tanx+ cot x= a, cosx + sinx =b.
D) eliminate x and y from a cot²x + b cot²y = 1, a cos²x + b cos²y = 1 and a sinx = b sin y. (a+ b)(a³ + b³)= ab(a+ b+1)(a+ b -1).
2) The angles of a triangle are in AP and the greatest angle is twice the smallest. Find the angles in radians.
3) If secx + tanx =√3 find secx - tanx . -5/4
4) Prove tan 130 tan140 = 1
5) If cosx = -4/3 find secx and cosecx.
6) Evaluate: tan 1. tan 2. Tan 3. ....tan 89. 1
7) If cosx - sinx =√2 sinx show cosx + sinx =√2 cosx.
8) If secx - tanx = p, find sinx.
9) if a cosx - b sinx =c express a sinx + b cosx in terms of a, b, c. ±√(a²+ b²- c²)
10) If cos⁴x + cos²x =1. Prove tan⁴x + tan²x = 1.
11) Find maximum and minimum value of 9 tan²x + 4 cot²x.
12) If x= r cos a cos b, y= r cos a sin b, z= r sin a, show x²+ y²+ z² = r².
13) Prove: (1- sinx)/(1+ sinx)= (sec x - tanx)².
14) If A, B, C, D are the angles of a cyclic quadrilateral, then prove cos A+ cos B+ cos C + cos D= 0.
15) If cos²x - sin²x = tan²y show that cos²y - sin²y = tan²x.
16) If x+ y= 90 and 2(cos²x - cos²y) = 1 then find the value of x. 30°
17) Value of 4(sin⁶x+ cos⁶x)- 6(sin⁴x + cos⁴x). -2
18) value of sec(-1680)sin 330. 1
19) If cosx + sinx =1, then find cosx - sinx. ±1
20) If tanx = -4/3 find sinx. ±4/5
21) The cotangent of the angles π/6, π/4, π/3 are in AP or G. P or HP or no sequences. GP
22) The equation (a+b)² sec²x = 4ab is possible when
a) a=0 b) b= 0 c) a= b. d) a≠ b
23) If z= cos²x + sec²x then the value always
a) z< 1 b) z= 1 c) 2>z>1 d) z≥2.
24) If sin²x + sinx = 1 then the value of cos¹²x + 3 cos¹⁰x + 3 cos⁸x + cos⁶x - 1. 0
25) Which of the following are true
a) tan 1°< tan 1 b) tan< tan 1°
. c) tan 1< tan 2
26) If tan⁴x + tan²x = 1. Show cos⁴x + cos²x = 1.
27) If sinx + cosecx = 2, show sin⁸x + cos⁸x = 2, also sinⁿx + cosⁿx = 2.
28) If x cos³a + y sin³s = sina cosa and x cos a - y sin a = 0 prove x² + y² = 1.
29) If tan x + sin x= m, tan x - sin x= n show m² - n² = √(4mn).
30) If x = cosec a - sin a and y= sec a - cos a then show x²y²(x² + y²+3) = 1.
31) If (sec a+ tan a)(sec b+ tan b) (sec c+ tan c)= (sec a- tan a)(sec b - tan b)(sec c+ tan c) then show each side = ±1.
32) If bx sin m = ay cos m and ax sec m - by cosec m = a² - b² show that x²/a² + y²/b² = 1.
33) If a sin x = b cos x = 2c {tanx/(1- tan²x)} show (a² - b²)² = 4c²(a² + b²).
34) If sin⁴(x/a) + cos⁴(x/b)= 1/(a+ b) prove sin⁸(x/a³) + cos⁸(x/b³)= 1/(a+ b)³.
35) If x= π/4n show sin²x + sin²3x + sin²5x + ..... to 2n terms = n.
36) If m² + m₁²+ 2mm₁ cos x = 1, n²+ n₁² + 2nn₁ cos x = 1 and mn + m₁n₁ + (mn₁ + m₁n) cos x= 0, then prove cosec²x = m²+ n².
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