SELINA Solutions for Class 10 Maths Chapter 21 - Trigonometrical Identities (Including Trigonometrical Ratios of Complementary Angles and Use of Four Figure Trigonometrical Tables)
Chapter 21 - Trigonometrical Identities (Including Trigonometrical Ratios of Complementary Angles and Use of Four Figure Trigonometrical Tables) Exercise Ex. 21(A)
Question 1
Prove:
Solution 1
Question 2
Prove:
Solution 2
Question 3
Prove:
Solution 3
Question 4
Prove:
Solution 4
Question 5
Prove:
Solution 5
Question 6
Prove:
Solution 6
Question 7
Prove:
Solution 7
Question 8
Prove:
Solution 8
Question 9
Prove:
Solution 9
Question 10
Prove:
Solution 10
Question 11
Prove:
Solution 11
Question 12
Prove:
Solution 12
Question 13
Prove:
Solution 13
Question 14
Prove:
Solution 14
Question 15
Prove:
Solution 15
Question 16
Prove:
Solution 16
Question 17
Prove:
Solution 17
Question 18
Prove:
Solution 18
Question 19
Prove:
Solution 19
Question 20
Prove:
Solution 20
Question 21
Prove:
Solution 21
Question 22
Prove:
Solution 22
Question 23
Prove:
Solution 23
Question 24
Prove:
Solution 24
Question 25
Prove:
Solution 25
Question 26
Prove:
Solution 26
Question 27
Prove:
Solution 27
Question 28
Prove:
Solution 28
Question 29
Prove:
Solution 29
Question 30
Prove:
Solution 30
Question 31
Prove:
Solution 31
Question 32
Prove:
Solution 32
Question 33
Prove:
Solution 33
Question 34
Solution 34
To prove:
Question 35
Prove:
Solution 35
Question 36
Prove:
Solution 36
Question 37
Prove:
Solution 37
Question 38
Prove:
Solution 38
Question 39
Prove:
Solution 39
Question 40
Prove:
Solution 40
Question 41
Prove:
Solution 41
Question 42
Prove:
Solution 42
Question 43
Prove:
Solution 43
Question 44
Prove:
Solution 44
Question 45
Prove:
Solution 45
Question 46
Prove:
Solution 46
Question 47
Prove:
Solution 47
Question 48
Prove:
Solution 48
Question 49
Prove:
Solution 49
Chapter 21 - Trigonometrical Identities (Including Trigonometrical Ratios of Complementary Angles and Use of Four Figure Trigonometrical Tables) Exercise Ex. 21(B)
Question 1
Prove that:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
(ix) 
Solution 1
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
Question 2
If 
= m and 
= n, then prove that x2
+ y2 = m2 + n2.
Solution 2
Question 3
If m= 
and n= 
, prove that 
Solution 3
Question 4
If 
, prove that 
Solution 4
Question 5
If 
and 
, prove that 
Solution 5
Given:
and
Question 6
If 
, prove that 
Solution 6
Question 7
If 
and 
, show that (m2 + n2)
cos2B = n2.
Solution 7
LHS = (m2 + n2) cos2B
Hence, (m2 + n2) cos2B = n2.
Chapter 21 - Trigonometrical Identities (Including Trigonometrical Ratios of Complementary Angles and Use of Four Figure Trigonometrical Tables) Exercise Ex. 21(C)
Question 1
Without using trigonometric tables, show that:
(i) 
(ii) 
(iii) 
Solution 1
(i)
(ii)
(iii)
Question 2
Express each of the following in terms of angles between 0°and 45°:
(i) sin 59°+ tan 63°
(ii) cosec 68°+ cot 72°
(iii)cos 74°+ sec 67°
Solution 2
Question 3
Show that:
(i) 
(ii) 
Solution 3
(i) 
(ii) 
Question 4
For triangle ABC, show that:
(i) 
(ii) 
Solution 4
(i) We know that for a triangle 
ABC
A + B + C = 180°
(ii) We know that for a triangle ABC
A + B + C = 180°
B + C = 180° - A
Question 5
Evaluate:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
(ix) 
Solution 5
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
Question 6
A triangle ABC is
right angled at B; find the value of 
Solution 6
Since, ABC is a right angled triangle, right angled at B.
So, A + C = 90
Question 7
Find (in each case, given below) the value of x if:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
Solution 7
(i)
Hence, x = 
(ii)
Hence, x = 
(iii)
Hence, x =
(iv)
Hence, x =
(v)
Hence, x = 
(vi)
Hence, x = 
(vii)
Hence, 
Question 8
In each case, given
below, find the value of angle A, where 
(i) 
(ii) 
Solution 8
(i)
(ii)
Question 9
Prove that:
(i) 
(ii) 
Solution 9
(i)
(ii)
Question 10
Evaluate :
Solution 10
Question 11
Evaluate
sin2 34° + sin2 56° + 2tan 18° tan72° - cot2 30°
Solution 11
Question 12
Without using trigonometrical tables, evaluate:
Solution 12
Chapter 21 - Trigonometrical Identities (Including Trigonometrical Ratios of Complementary Angles and Use of Four Figure Trigonometrical Tables) Exercise Ex. 21(D)
Question 1
Use tables to find sine of:
(i) 21°
(ii) 34° 42'
(iii) 47° 32'
(iv) 62° 57'
(v) 10° 20' + 20° 45'
Solution 1
(i) sin 21o = 0.3584
(ii) sin 34o 42'= 0.5693
(iii) sin 47o 32'= sin (47o 30' + 2') =0.7373 + 0.0004 = 0.7377
(iv) sin 62o 57' = sin (62o 54' + 3') = 0.8902 + 0.0004 = 0.8906
(v) sin (10o 20' + 20o 45') = sin 30o65' = sin 31o5' = 0.5150 + 0.0012 = 0.5162
Question 2
Use tables to find cosine of:
(i) 2° 4’
(ii) 8° 12’
(iii) 26° 32’
(iv) 65° 41’
(v) 9° 23’ + 15° 54’
Solution 2
(i) cos 2° 4’ = 0.9994 - 0.0001 = 0.9993
(ii) cos 8° 12’ = cos 0.9898
(iii) cos 26° 32’ = cos (26° 30’ + 2’) = 0.8949 - 0.0003 = 0.8946
(iv) cos 65° 41’ = cos (65° 36’ + 5’) = 0.4131 -0.0013 = 0.4118
(v) cos (9° 23’ + 15° 54’) = cos 24° 77’ = cos 25° 17’ = cos (25° 12’ + 5’) = 0.9048 - 0.0006 = 0.9042
Question 3
Use trigonometrical tables to find tangent of:
(i) 37°
(ii) 42° 18'
(iii) 17° 27'
Solution 3
(i) tan 37o = 0.7536
(ii) tan 42o 18' = 0.9099
(iii) tan 17o 27' = tan (17o 24' + 3') = 0.3134 + 0.0010 = 0.3144
Question 4
Use tables to find the acute angle 
, if the value of sin is:
(i) 0.4848
(ii) 0.3827
(iii) 0.6525
Solution 4
(i) From the tables, it is clear that sin 29o = 0.4848
Hence, = 29o
(ii) From the tables, it is clear that sin 22o 30' = 0.3827
Hence, = 22o 30'
(iii) From the tables, it is clear that sin 40o 42' = 0.6521
sin - sin 40o 42' = 0.6525 -; 0.6521 = 0.0004
From the tables, diff of 2' = 0.0004
Hence, = 40o 42' + 2' = 40o 44'
Question 5
Use tables to find the acute angle 
, if the value of cos is:
(i) 0.9848
(ii) 0.9574
(iii) 0.6885
Solution 5
(i) From the tables, it is clear that cos 10° = 0.9848
Hence, = 10°
(ii) From the tables, it is clear that cos 16° 48’ = 0.9573
cos - cos 16° 48’ = 0.9574 - 0.9573 = 0.0001
From the tables, diff of 1’ = 0.0001
Hence, = 16° 48’ - 1’ = 16° 47’
(iii) From the tables, it is clear that cos 46° 30’ = 0.6884
cos q - cos 46° 30’ = 0.6885 - 0.6884 = 0.0001
From the tables, diff of 1’ = 0.0002
Hence, = 46° 30’ - 1’ = 46° 29’
Question 6
Use tables to find the acute angle 
, if the value of tan q is:
(i) 0.2419
(ii) 0.4741
(iii) 0.7391
Solution 6
(i) From the tables, it is clear that tan 13° 36’ = 0.2419
Hence, = 13° 36’
(ii) From the tables, it is clear that tan 25° 18’ = 0.4727
tan - tan 25° 18’ = 0.4741 - 0.4727 = 0.0014
From the tables, diff of 4’ = 0.0014
Hence, = 25° 18’ + 4’ = 25° 22’
(iii) From the tables, it is clear that tan 36° 24’ = 0.7373
tan - tan 36° 24’ = 0.7391 - 0.7373 = 0.0018
From the tables, diff of 4’ = 0.0018
Hence, = 36° 24’ + 4’ = 36° 28’
Chapter 21 - Trigonometrical Identities (Including Trigonometrical Ratios of Complementary Angles and Use of Four Figure Trigonometrical Tables) Exercise Ex. 21(E)
Question 1
Prove the following identities:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
(ix) 
(x) 
(xi) 
(xii) 
(xiii) 
(xiv) 
(xv) 
(xvi) 
(xvii) 
Solution 1
(i) 
(ii)
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
(ix) 
(x) 
(xi)
(xii) 
(xiii) 
(xiv) 
(xv) 
(xvi) 
(xvii) 
Question 2
If 
and 
, then prove that:
q(p2 - 1) = 2p
Solution 2
Question 3
If 
, show that:
Solution 3
Question 4
If 
, show that:
Solution 4
Question 5
If tan A = n tan B and sin A = m sin B, prove that:
Solution 5
Question 6
(i) If 2 sinA - 1 = 0, show that:
sin 3A = 3 sinA - 4 sin3A
(ii) If 4 cos2A - 3 = 0, show that:
cos 3A = 4 cos3A - 3 cosA
Solution 6
(i) 2 sinA - 1 = 0
(ii)
Question 7
Evaluate:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
Solution 7
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
Question 8
Prove that:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Solution 8
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Question 9
If A and B are complementary angles, prove that:
(i) 
(ii) 
(iii) cosec2A + cosec2B = cosec2A cosec2B
(iv) 
Solution 9
Since, A and B are complementary angles, A + B = 90°
(i)
(ii)
(iii)
= cosec2A [sec(90 - B)]2
= cosec2A cosec2B
(iv) 
Question 10
Prove that:
Solution 10
Question 11
If 4cos2A - 3 = 0 and 0°
A 90°, then prove that:
(i) sin3A= 3 sinA - 4 sin3A
(ii) cos3A= 4 cos3A - 3 cosA
Solution 11
4 cos2A - 3 = 0
Question 12
Find A, if 0°A 90° and:
(i) 
(ii) sin 3A - 1 = 0
(iii) 
(iv) 
(v) 
Solution 12
(i)
(ii) sin 3A - 1 = 0
(iii)
(iv)
(v)
Question 13
If 0° < A < 90°; find A, if:
(i) 
(ii) 
Solution 13
(i)
(ii)
Question 14
Prove that:
(cosec A - sin A) (sec A - cos A) sec2A = tan A
Solution 14
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Question 15
Prove the identity (sin θ + cos θ) (tan θ + cot θ) = sec θ + cosec θ.
Solution 15
Question 16
Evaluate without using trigonometric tables,
sin2 28° + sin2 62° + tan2 38° - cot2 52° + 
sec2
30°
Solution 16
sin2
28° + sin2
62° + tan2
38° - cot2
52° + sec2 30°
= sin2
28° + [sin (90 -
28)°]2 +
tan2 38° - [cot(90 - 38)°]2 +
sec2 30°
= sin2
28° + cos2 28° + tan2
38° - tan2
38° + sec2 30°
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