Class: XI
Trigonometry
Measurement of Angles
i)
The angles of a triangle are in A.P. The number of degrees in the least is to the number
of radians in the greatest as 60:. Find the angles in degrees.
Ans. 30O, 60O, 90O
ii)
A circular wire of radius 7.5cm is cut and bent so as to lie along the circumference of a
hoop whose radius is 120cm. Find in degrees the angle, which is subtended at the centre
of the hoop.
Ans. 22O30'
iii)
Find the angle between the minute hand of a clock and the hour hand when the time is
7:20 A.M.
Ans. 100O
iv)
Find in degrees and radians the angle between the hour hand and the minute hand of a
clock at half past three.
Ans. 5 rad , 75O
12
Trigonometric Functions
(A)
Signs of Trigonometric Function.
i)
Find the values of Cos  and tan , if Sin 
ii)
= –3 and     3
Ans. – 4 , 3
5
2
5 4
Find the values of other Trigonometric Ratios, if Sin  = – 26 &  lies in quadrant III.
5
Ans. Cos  = -1 , tan  = 26, Cosec  = -5 , Sec  = -5, Cot  = 1
5
26
26
iii)
If Cos  = -1 and     3 , find the value of 4tan2  – 3Cosec2 .
2
2
Ans. 8
iv)
If Sec  = 2 and 3   2. Find the value of 1 + tan  + cosec 
2
1 + cot  – Cosec 
Ans. – 1
(B)
Values of Trigonometric Ratios
i)
Find the value of the following:
a) Sin 315O
b) Cos 210O
O
e) Cosec 390
f) Cot 570O
c) Cos (-480O)
g) tan 480O
d) Sin(-1125O)
h) Cos 270O
ii)
Prove that Cos 510O Cos 330O + Sin 390O Cos 120O = -1.
iii)
Prove that Sin (-420O) Cos390O + Cos (-660O) Sin 330O = -1.
iv)
Prove that
v)
If A, B, C, D are angles of a cyclic quadrilateral, Prove that
Cos A + Cos B + Cos C + Cos D = 0.
vi)
In any quadrilateral ABCD, prove that
Sin (A+B) + Sin (C+D) = 0 & Cos (A+B) = Cos (C+D)
Cos (90O + ) Sec (-) tan (180O - )
Sec (360O - ) Sin (180O + ) Cot (90O - )
-1
Trigonometric Ratios of Compound Angles
i)
If Cos  = 13 & Cos  = 1 where ,  are acute angles show that  -  = 
14
7
3
ii)
If tan  = m
& tan  = 1
prove that  +  = 
m+1
2m + 1
4
iii)
If A + B =  , prove that (Cot A – 1) (Cot B – 1) = 2.
4
iv)
Sin2 A = Cos2 (A – B) + Cos2 B – 2Cos (A – B) Cos A Cos B.
v)
Prove Sin2 
8
vi)
vii)
If Cos ( + ) = 4 ,Sin ( - ) = 5 and  ,  lie between 0 and  , prove that tan 2 = 56
5
13
4
33
If Cot  = 1 , Sec  = -5 where  <  < 3 and  <  < . Find the value of tan (+).
2
3
2
2
State the quadrant in which + terminates.
Ans. 1st quadrant.
viii)
tan 70O = tan 20O + 2tan 50O.
ix)
If tan ( + ) = n tan ( - ), show that (n + 1) Sin2 = (n – 1) Sin 2.
x)
If sin +sin  = a & cos +cos  = b, show that cos ( + ) = b2–a2 & sin (+) = 2ab
b2+a2
a2 + b2
xi)
If tan ( cos ) = cot ( sin ), prove that cos
xii)
If cos ( - ) + cos ( - ) + cos ( - ) = -3 ,
2
Prove that cos  + cos  + cos  = sin  + sin  + sin  = 0
xiii)
If  and  are the solutions of the equation a tan  + b sec  = c then
Show that tan ( + ) = 2ac
a2 – c2
If  and  are the solutions of a cos  + b sin  = c,
Prove that cos ( - ) = 2c2 – (a2 + b2)
a2 + b2
xiv)
A - Sin2 
2
8
A
2
1 Sin A.
2
 – =  1
4
22
Transformation Formulae
i)
Convert the following as sum or difference of sines and cosines
a)
Sin 75O Cos 15O
Ans. 1 (Sin90O + Sin 60O)
2
b)
2cos 4 cos 3
Ans. cos 7 + cos 
ii)
Prove that cos 20O cos 40O cos 60O cos 80O = 1
16
iii)
Prove that sin 20O sin 40O sin 60O sin 80O = 3
16
iv)
Prove that tan 20O tan 40O tan 80O = tan 60O
v)
Prove that sin  + sin  + 2 + sin  + 4 = 0.
3
3
vi)
cos  + cos  + cos  + cos ( +  + ) = 4 cos  + 
2
vii)
Show that
Sin 8 Cos 23 + Cos 13 Sin 35
3
6
3
6
cos  + 
2
1
2
Trigonometric Ratios of Multiple and Submultiples Angles
i)
Prove that 1 + Sin  – Cos 
1 + Sin  + Cos 
ii)
Prove that
Cos 2
1 + Sin 2
tan 
2
tan  – 
4
cos  + 
2
2+
2+
2 + 2 cos 8 = 2 cos 
iii)
Prove that
iv)
Sec 8 – 1
Sec 4 – 1
v)
Cos4  + Cos4 3 + Cos4 5 + Cos4 7
8
8
8
8
vi)
Sin4  + Sin4 3 + Sin4 5 + Sin4 7
8
8
8
8
vii)
Cos 5A = 16Cos5 A – 20Cos3 A + 5Cos A
viii)
Sin A Sin (60 – A) Sin (60 + A) = 1 Sin 3A
4
ix)
Find the values of Cos 22½O, Sin 22½O, tan 22½O, Sin 7½O, cos 7½O, cot 7½O, tan 11¼O.
tan 8
tan 2
3
2
3
2
Trigonometric Ratios of Some Important Angles
i)
Sin 18O, Cos 18O, Sin 36O, Cos 36O.
ii)
Prove that Sin2 24O – Sin2 6O = 5 – 1
8
iii)
Prove that Sin  . Sin 13
10
10
iv)
Prove that Sin  Sin 2 Sin 3 Sin 4
5
5
5
5
v)
Prove that 16Cos 2 Cos 4 Cos 8 Cos 16
15
15
15
15
1
vi)
Prove that 1 + cos  1 + cos 3
10
10
1 + cos 9
10
vii)
tan 6O tan 42O tan 66O tan 78O = 1
–1
4
5
16
1 + cos 7
10
Trigonometric Equations
i) sin x tan x – 1 = tan x – sin x
Ans x  n   1
ii) cot x + tan x = 2 cosec x
Ans
iii) tan x + tan 2x + √3 tan x tan 2x = √3 Ans
iv) tan3x – 3 tan x = 0
Ans
v) sin 2x + sin 4x + sin 6x = 0
Ans
n

1
16
or x  m 
2

x = 2nπ ± , n  Z
3
n 
x
 ,n Z
3
9

2
x  n or n 
or n 
,nZ
3
3
n

x 
, x  m 
, n, m  
4
3
Trigonometric equations of the form a cos + b sin = c wherec  (a2 + b2)
i) 3cos  + sin  = 2
ii) 2 sec  + tan  = 1
iii) Cot  + cosec  = 3
iv) 3 cos  + sin  = 1
v) Cosec  = 1 + cot 
3
, m, n  Z
4