BADHANEDUCATION D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI
CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT
CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315
1. Find the radian measures corresponding to the following degree measures:
(i) 250 (ii)-47030’ (iii)2400 (iv)5200
w
Solution:
Radians = 180 degrees
•
du
ne
ha 5
1
ad 3
.b 44
w 1
0
w
81
09
1 degree =
(i) 25° = 25 x
radians
=
Radians.
(ii) -47° 30’ = -47° 30’ x
(iii) 240° = 240 x
= - 0.2638
=
(iv) 520° = 520 x
Radians.
Radians
=
Radians
2. Find the degree measures corresponding to the following radian measures
(use π=
i)
(ii)-4 (iii)
(iv)
Solution:
Radians = 180°
∴ 1 Radian =
degrees
ii) 1 radian =
× -4 =
-4radian
= -32.72 × 7 = -229o5'24''
.in
on
= 39.375º
=
ti
ca
•
).(i)
iii) 1 radian =
=
= 3000
1
CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA,
DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in DELHI AND OTHER MAJOR CITIES
EMAIL:1 [email protected] web site www.badhaneducation.in
BADHANEDUCATION D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI
CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT
CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315
iv) 1 radian =
= 210o
w
09
du
ne
ha 5
1
ad 3
.b 44
w 1
0
w
81
3. A wheel makes 360 revolutions in one minute. Through how many radians does
it turn is one second?
•
Solution:
One complete revolution = 2π
360 revolutions = 360 × 2π€= 720π radians / minute = 720π /60 radians/sec =
12π radians/sec.
4. In a circle of diameter 40cm. The length of a chord is 20cm. Find the length of
minor arc corresponding to the chord.
•
Solution:
ca
.
ti
5. If, in two circles, arcs of the same length subtend angles of 60o and 75o at the
centre, find the ratio of their radii.
on
•
Solution:
Let the radii be r1 and r2.
Let the angles subtend by the arcs in two circles be θ1 and θ2.
l = θ r where θ is the angle, l the length of an arc and r the radius of the circle.
=
θ2 = 75o = 75 ×
=
r1 =
radian
radian
.in
θ1 = 60o = 60 ×
r2
2
CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA,
DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in DELHI AND OTHER MAJOR CITIES
EMAIL:2 [email protected] web site www.badhaneducation.in
BADHANEDUCATION D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI
CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT
CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315
r1 / r2 =
=
Therefore the required ratio is 5:4.
w
09
du
ne
ha 5
1
ad 3
.b 44
w 1
0
w
81
6. Find the angle in radian through which a pendulum swings if its length is 75cm
and the tip described an arcs of length
(i) 10cm
(ii) 15cm
(iii) 21cm
•
Solution:
(i) Length of the pendulum (r) = 75cm
Length of an arc (l) = 10cm
θ = l/r = 10/75radians = 2/15radians
(ii) Length of the pendulum (r) = 75cm
Length of an arc (l) = 15cm
θ = l/r = 15/75 radians = 1/5radians
(iii) Length of the pendulum (r) = 75cm
Length of an arc (l) = 21cm
θ = l/r = 21/75 radians = 7/25radians
7. Find the values of other five trigonometric functions
1. Cos x =
•
, x lies in third quadrant.
Solution:
, cosec x = -
, tanx =
, cotx =
ca
sin x =
secx = -2
8. Find the values of other five trigonometric functions
, x lies in second quadrant.
Solution:
, secx = -
9. Cot x =
•
, tanx = -
, cotx = -
,Cosecx =
, x lies in third quadrant.
.in
cos x = -
on
•
ti
(i) Sin x =
Solution:
Tanx =
, sinx = -
, Cosec x = -
, cosx = -
, secx = -
3
CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA,
DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in DELHI AND OTHER MAJOR CITIES
EMAIL:3 [email protected] web site www.badhaneducation.in
BADHANEDUCATION D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI
CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT
CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315
10. Find the values of the other five trigonometric functions in the following
problem: sec θ = 13/5, θ lies in fourth quadrant.
•
du
ne
ha 5
1
ad 3
.b 44
w 1
0
w
81
w
Solution:
sin θ , tan θ , cosec θ and cot θ are negative in IV th quadrant sec θ and cos θ are positive in
IVth quadrant.
sec θ = 13/5
cos θ = 1/sec θ = 5/13
09
sin2 θ = 1∴ sin θ =
cosec θ =
∴ tan θ =
=
cot θ =
=
11. Tan x =
•
, x lies in second quadrant.
Solution:
, sinx =
, cosecx =
, cosx = -
ca
cotx = -
, secx = -
Solution:
sin 765o = sin
= sin
= sin
=
13. Find the value of the following trigonometric function: cosec(-1410o)
•
Solution:
cosec(-1410o) = -cosec 1410o = -cosec (8π – 30o) = cosec 30o = 2
14. Find the value of the following trigonometric function:sin (
.in
on
•
ti
12. Find the values of the trigonometric functions sin 765o
)
4
CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA,
DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in DELHI AND OTHER MAJOR CITIES
EMAIL:4 [email protected] web site www.badhaneducation.in
BADHANEDUCATION D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI
CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT
CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315
Solution:
•
Sin
= -Sin
w
15. 2sin2
+2cos2
= -Sin(660)= - Sin (2x360 – 60) = -(-sin60) = sin60 =
+2sec2
= 10
du
ne
ha 5
1
ad 3
.b 44
w 1
0
w
81
Solution:
•
09
L.H.S = 2sin2
+2cos2
= 2sin2
= 2sin2
=
=
=
=
+2sec2
+2cos2
+ 2cos2
2×
+ 2×
1 + 1+ 8
10
R.H.S
+2sec2
+ 2sec2
+ 2× 4
16. Find the value of:(i) sin 750 (ii) tan150
•
Solution:
i)Sin (30+45)
= sin30cos45+cos30sin45 Sin(A+B)
=SinACosB+CosAsinB
=
ca
=
ti
(ii)tan15 = tan(45 – 30)
on
=
.in
=
=
17. Prove the following:Cos
cos
- sin
Sin
= sin(x+y)
5
CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA,
DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in DELHI AND OTHER MAJOR CITIES
EMAIL:5 [email protected] web site www.badhaneducation.in
BADHANEDUCATION D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI
CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT
CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315
•
Solution:
This is of the form CosACosB-SinASinB = Cos(A+B)
= Cos
09
du
ne
ha 5
1
ad 3
.b 44
w 1
0
w
81
w
∴ L H S = Cos
= Sin(x+y)
Hence proved.
= cot2x
18.
•
Solution:
=
= -cos2x/-sin2x
= cot2x
19. cos
cos(2 +x)
=1
Solution:
•
= sinxcosx
=Sinx.cosx
20. sin (n + 1)x sin (n + 2)x + cos(n + 1)xcos(n + 2)x = cosx.
21. cos
=-
sinx
.in
Solution:
L.H.S = sin (n + 1)x sin (n + 2)x + cos(n + 1)xcos(n + 2)x
= cos[(n + 1)x - (n + 2)x]
= cos (-x)
= cosx
= RH.S
on
•
ti
ca
=sinx.cosx
=sin2x+cos2x = 1
(Hence Proved)
Solution:
•
cos
cosx – sin
sinx-
6
CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA,
DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in DELHI AND OTHER MAJOR CITIES
EMAIL:6 [email protected] web site www.badhaneducation.in
BADHANEDUCATION D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI
CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT
CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315
= cos
cosx – sin
= -2sin
sinx
sinx -
w
=-
du
ne
ha 5
1
ad 3
.b 44
w 1
0
w
81
09
22. sin2 6x – sin2 4x = sin2x sin10x
•
Solution:
L.H.S = sin2 6x – sin2 4x
[Using the formulae 2sin2x = 1 - cos2x]
=
=-
(cos12x – cos8x)
=[cos(10x + 2x) – cos(10x - 2x)]
[Using the formulae -2sinθ sinφ = cos(θ + φ ) - cos(θ - φ )]
=
× 2 sin10x sin2x
= sin10x sin2x
23. cos22x – cos26x = sin4x sin8x
•
Solution:
L.H.S = cos22x – cos26x
[Using the formulae 2cos2x = 1+ cos2x]
=-
(cos12x – cos4x)
•
.in
24. sin2x + 2sin4x + sin6x = 4cos2x sin4x
on
= × 2 sin8x sin4x
= sin4x sin8x
ti
=[cos(8x + 4x) – cos(8x - 4x)]
[Using the formulae -2sinθ sinφ = cos(θ + φ ) - cos(θ - φ )]
ca
=
Solution:
L.H.S = sin2x + 2sin4x + sin6x
=
=
=
=
=
2sin
cos
+ 2sin4x
2sin4xcos2x + 2sin4x
2sin4x(cos2x + 1)
4sin4xcos2x
R.H.S
7
CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA,
DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in DELHI AND OTHER MAJOR CITIES
EMAIL:7 [email protected] web site www.badhaneducation.in
BADHANEDUCATION D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI
CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT
CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315
25. cot4x(sin5x + sin3x) = cotx(sin5x – sin3x)
•
Solution:
L.H.S = cot4x(sin5x + sin3x)
w
[Using the formula sin5x + sin3x = 2 sin
cos
]
du
ne
ha 5
1
ad 3
.b 44
w 1
0
w
81
= 2cot4x sin
cos
09
= 2cot4x sin
cos
= 2cot4x sin4x cosx
= 2cos4x cosx
R.H.S = cotx(sin5x – sin3x)
[Using the formula sin5x - sin3x = 2 cos
= 2cotx cos
sin
]
sin
= 2cotx cos
sin
= 2cos4x cosx
∴ L.H.S = R.H.S
26. Prove that
•
=
Solution:
L.H.S =
ca
=
ti
on
=
=
.in
=
= R.H.S
27. Prove that
•
= tan4x
Solution:
L.H.S =
8
CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA,
DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in DELHI AND OTHER MAJOR CITIES
EMAIL:8 [email protected] web site www.badhaneducation.in
BADHANEDUCATION D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI
CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT
CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315
=
w
=
du
ne
ha 5
1
ad 3
.b 44
w 1
0
w
81
09
=
= tan4x
= R.H.S
28. Prove that
= tan
Solution:
•
L.H.S =
=
=
= tan
= R.H.S
ca
29.
ti
Solution:
•
on
=
.in
=
= tan2x.
30. Prove that
•
=2sinx
Solution:
L.H.S =
9
CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA,
DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in DELHI AND OTHER MAJOR CITIES
EMAIL:9 [email protected] web site www.badhaneducation.in
BADHANEDUCATION D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI
CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT
CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315
=
=
w
[Using the identity cos2x = cos2x – sin2x and sinx – siny = 2cos
sin
]
du
ne
ha 5
1
ad 3
.b 44
w 1
0
w
81
09
=
= 2sinx
= R.H.S
31. Prove that
Solution:
•
=
=
cot3x
=
R.H.S
Hence proved
•
Solution:
Cot2x(cotx – cot 3x)- cot 3x.cotx
ti
ca
32. cot x cot 2x – cot 2x cot3x – cot3x cotx =1
= cot 3x.cotx
.in
= cot2x
- cot3x.cotx
on
= cot 2x
=
=
10
CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA,
DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in DELHI AND OTHER MAJOR CITIES
EMAIL:10 [email protected] web site www.badhaneducation.in
BADHANEDUCATION D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI
CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT
CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315
=
du
ne
ha 5
1
ad 3
.b 44
w 1
0
w
81
w
=
09
=
=
=
=
=
ca
=
ti
=
=
.in
on
=
= 1.
33. tan4x =
Solution:
•
L.H.S
= tan4x
11
CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA,
DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in DELHI AND OTHER MAJOR CITIES
EMAIL:11 [email protected] web site www.badhaneducation.in
BADHANEDUCATION D96 MUNIRKA VILLAGE NEW DELHI 110067 & VIKAS PURI NEW DELHI
CONTACT FOR COACHING MATHEMATICS FOR 11TH 12TH NDA DIPLOMA SSC CAT SAT CPT
CONTACT FOR ADMISSION GUIDANCE B.TECH BBA BCA, MCA MBA DIPLOMA AND OTHER COURSES 09810144315
= tan(3x+x)
=
du
ne
ha 5
1
ad 3
.b 44
w 1
0
w
81
w
09
=
=
=
=
L.H.S = R.H.S
34. Prove that cos4x = 1 - 8sin2xcos2x.
ti
Solution:
= cos4x = 1-2sin22x
= 1-2 (2sinx cosx)2
= 1-8 sin2x cos2x = R.H.S
Hence proved.
ca
•
•
= 4(2cos2x-1)3 - 3(2cos2x-1)
= 4[(2cos2x)3 - 3×(2cos2x)2×1 + 3 × 2 cos2x×1 - 13]-6cos2x + 3
.in
Solution:
cos6x = 4 cos32x - 3 cos2x
on
35. Prove that cos6x = 32cos6 x- 48cos4x + 18cos2x-1
= 32cos6x - 48cos4x + 18cos2x -1
= R.H.S
Hence proved
12
CONTACT FOR MATHEMATICS GROUP/HOME COACHING FOR CLASS 11TH 12TH BBA,BCA,
DIPLOMA CPT CAT SSC AND OTHER EXAMS Also home tutors for other subjects in DELHI AND OTHER MAJOR CITIES
EMAIL:12 [email protected] web site www.badhaneducation.in