## Chapter 27 - Trigonometrical Ratios of Standard Angles Exercise Ex. 27.1

Question 1
Solution 1
Question 2
Solution 2

Question 3
Prove that :
sin 60° .cos 30° - sin 60°. sin 30° =
Solution 3
Question 4
Prove that :
cos 60° . cos 30° - sin 60° . sin 30° = 0
Solution 4
Question 5
Prove that :
sec245° - tan245° = 1
Solution 5
Question 6
Prove that :
Solution 6
Question 7
Find the value of 'A', if
2 cos A = 1
Solution 7
Question 8
Find the value of 'A', if
2 sin 2A = 1
Solution 8
Question 9
Find the value of 'A', if
Solution 9
Question 10
Find the value of 'A', if
2 cos 3A = 1
Solution 10
Question 11
Find the value of 'A', if
Solution 11
Question 12
Find the value of 'A', if
cot 3A = 1
Solution 12
Question 13
Find the value of 'A', if
(1 - cosec A)(2 - sec A) = 0
Solution 13
Question 14
Find the value of 'A', if
(2 - cosec 2A) cos 3A = 0
Solution 14
Question 15
If sin α + cosβ = 1 and α= 90°, find the value of 'β'.
Solution 15
Question 16
Solve for 'Ө':
Solution 16
Question 17
Solve for 'Ө':
cot2(Ө - 5)° = 3
Solution 17
Question 18
Solve for 'Ө':
Solution 18
Question 19
Solution 19

Question 20
If sin Ө = cosӨ and 0° < Ө<90°, find the value of 'Ө'.
Solution 20
Question 21
If tan Ө= cot Ө and 0°≤Ө≤ 90°, find the value of 'Ө'.
Solution 21
Question 22
Solution 22
Question 23
If Ө = 30°, verify that:
Solution 23
Question 24
If Ө = 30°, verify that:
Solution 24
Question 25
If A = 30°, verify that:
Solution 25
Question 26
If Ө = 30°, verify that:
sin 3Ө = 4sin Ө.sin(60° - Ө) sin(60° + Ө)
Solution 26
Question 27
If Ө = 30°, verify that:
1 - sin 2Ө= (sin Ө - cosӨ)2
Solution 27
Question 28
Evaluate the following:
Solution 28
Question 29
Evaluate the following:
Solution 29
Question 30
If Ө = 15°, find the value of:
cos 3Ө - sin 6Ө + 3sin (5Ө + 15°) - 2 tan2
Solution 30
Question 31
If A = 30° and B = 60°, verify that:
sin (A + B) = sin A cos B + cos A sin B
Solution 31
Question 32
If A = 30° and B = 60°, verify that:
cos (A + B) = cos A cos B - sin A sin B
Solution 32
Question 33
If A = 30° and B = 60°, verify that:
Solution 33
Question 34
If A = 30° and B = 60°, verify that:
Solution 34
Question 35
If A = B = 45°, verify that
sin (A - B) = sin A .cos B - cos A.sin B
Solution 35
Question 36
If A = B = 45°, verify that
cos (A - B) = cosA.cos B + sin A.sin B
Solution 36
Question 37
Solution 37
Question 38
If Ө < 90°, find the value of:
sin2Ө + cos2Ө
Solution 38
Question 39
Solution 39

Question 40
If Ө < 90°, find the value of:
Solution 40
Question 41
Ifsec 2Ө = 2 and Ө< 90°, find the value of
Ө
Solution 41
Question 42
Ifsec 2Ө = 2 and Ө< 90°, find the value of
cos 3Ө
Solution 42
Question 43
Ifsec 2Ө = 2 and Ө< 90°, find the value of
cos2 (30° + Ө) + sin2 (45° - Ө)
Solution 43
Question 44
In the given figure, PQ = 6 cm, RQ = x cm and RP = 10 cm, find
a. cosӨ
b. sin2 Ө - cos2Ө
c. Use tan Ө to find the value of RQ
Solution 44
Question 45
Find the value of:
If 3 tan2Ө - 1 = 0, find the value
a. cos 2Ө
b. sin 2Ө
Solution 45
Question 46
Solution 46
Question 47
Solution 47
Question 48
Solution 48
Question 49
Solution 49

Question 50
Solution 50

## Chapter 27 - Trigonometrical Ratios of Standard Angles Exercise Ex. 27.2

Question 1
Find the value of 'x' in each of the following:
Solution 1

Question 2
Find the value of 'x' in each of the following:
Solution 2

Question 3
Find the value of 'x' in each of the following:
Solution 3
Question 4
Find the value of 'x' in each of the following:
Solution 4

Question 5
Find the length of AD.
Given: ∠ABC = 60°, ∠DBC = 45° and BC = 24 cm.
Solution 5

Question 6
Find lengths of diagonals AC and BD. Given AB = 24 cm and ∠BAD = 60°.
Solution 6
Question 7
In a trapezium ABCD, as shown, AB‖ DC, AD = DC = BC = 24 cm and ∠A = 30°. Find:
length of AB

Solution 7

Construction: Draw DP ⊥ AB and CM ⊥ AB

Question 8
Find the length of EC.

Solution 8
Question 9
In the given figure, AB and EC are parallel to each other. Sides AD and BC are 1.5 cm each and are perpendicular to AB. Given that ∠AED = 45° and ∠ACD = 30°. Find:
a. AB
b. AC
c. AE
Solution 9

Question 10
In the given figure, ∠B = 60°, ∠C = 30°, AB = 8 cm and BC = 24 cm. Find:
a. BE
b. AC
Solution 10

Question 11
Find:
a. BC
b. AD
c. AC
Solution 11
Question 12
Find the value 'x', if:
Solution 12

Question 13
Find the value 'x', if:
Solution 13

Question 14
Find the value 'x', if:
Solution 14
Question 15
Find the value 'x', if:
Solution 15

Question 16
Find the value 'x', if:
Solution 16

Question 17
Find the value 'x', if:
Solution 17
Question 18

Solution 18
Construction: Draw BX ⊥ AE
Then, BD = EX = 14 cm and BX = ED
AX = AE - EX = 16 - 14 = 2

Question 19

Solution 19

Question 20

Solution 20

Question 21
Find x and y, in each of the following figure:

Solution 21

Question 22
Find x and y, in each of the following figure:

Solution 22
Question 23
Solution 23

Question 24
Solution 24

Question 25
Solution 25
2
Question 26
Solution 26

Question 27
In right-angled triangle ABC; B = 90°. Find the magnitude of angle A, if:
Solution 27
Consider the following figure,

Question 28
A ladder is placed against a vertical tower. If the ladder makes an angle of 30° with the ground and reaches upto a height of 18 m of the tower; find length of the ladder.
Solution 28

Question 29
The perimeter of a rhombus is 100 cm and obtuse angle of it is 120°. Find the lengths of its diagonals.
Solution 29
Consider the following figure,

Question 30
In the given figure; ∠B = 90°, ∠ADB = 30°, ∠ACB = 45° and AB = 24 m.
Find the length of CD.
Solution 30

Question 31
In the given figure, a rocket is fired vertically upwards from its launching pad P. It first rises 20 km vertically upwards and then 20 km at 60° to the vertical. PQ represents the first stage of the journey and QR the second. S is a point vertically below R on the horizontal level as P, find:
a. the height of the rocket when it is at point R.
b. the horizontal distance of point S from P.
Solution 31
Draw QM ⊥ RS.
Clearly, ∠RQM = 30°

## Chapter 27 - Trigonometrical Ratios of Standard Angles Exercise Ex. 27.3

Question 1
Evaluate the following:
Solution 1
Question 2
Evaluate the following:
Solution 2
Question 3
Evaluate the following:
Solution 3
Question 4
Evaluate the following:
Solution 4
Question 5
Evaluate the following:
Solution 5
Question 6
Evaluate the following:
Solution 6
Question 7
Evaluate the following:
sin 31° - cos 59°
Solution 7
Question 8
Evaluate the following:
cot 27° - tan 63°
Solution 8
Question 9
Evaluate the following:
cosec 54° - sec 36°
Solution 9
Question 10
Evaluate the following:
sin 28° sec 62° + tan 49° tan 41°
Solution 10
Question 11
Evaluate the following:
sec 16° tan 28° - cot 62° cosec 74°
Solution 11
Question 12
Evaluate the following:
sin 22° cos 44° - sin 46° cos 68°
Solution 12
Question 13
Evaluate the following:
Solution 13
Question 14
Evaluate the following:
Solution 14
Question 15
Evaluate the following:
Solution 15
Question 16
Evaluate the following:
Solution 16
Question 17
Express each of the following in terms of trigonometric ratios of angles between 0° and 45°:
sin 65° + cot 59°
Solution 17
Question 18
Express each of the following in terms of trigonometric ratios of angles between 0° and 45°:
cos 72° - cos 88°
Solution 18
Question 19
Express each of the following in terms of trigonometric ratios of angles between 0° and 45°:
cosec 64° + sec 70°
Solution 19
Question 20
Express each of the following in terms of trigonometric ratios of angles between 0° and 45°:
tan 77° - cot 63° + sin 57°
Solution 20
Question 21
Express each of the following in terms of trigonometric ratios of angles between 0° and 45°:
sin 53° + sec 66° - sin 50°
Solution 21
Question 22
Express each of the following in terms of trigonometric ratios of angles between 0° and 45°:
cos 84° + cosec 69° - cot 68°
Solution 22
Question 23
Evaluate the following:
sin 35° sin 45° sec 55° sec 45°
Solution 23
Question 24
Evaluate the following:
cot 20° cot 40° cot 45° cot 50° cot 70°
Solution 24
Question 25
Evaluate the following:
cos 39° cos 48° cos 60° cosec 42° cosec 51°
Solution 25
Question 26
Evaluate the following:
sin (35° + Ө) - cos (55° - Ө) - tan (42° + Ө) + cot (48° - Ө)
Solution 26
Question 27
Evaluate the following:
tan (78° + Ө) + cosec (42° + Ө) - cot (12° - Ө) - sec(48° - Ө)
Solution 27
Question 28
Evaluate the following:
Solution 28
Question 29
Evaluate the following:
Solution 29
Question 30
Evaluate the following:
Solution 30
Question 31
Evaluate the following:
Solution 31
Question 32
Evaluate the following:
Solution 32
Question 33
If cos 3Ө = sin (Ө - 34°), find the value of Ө if 3Ө is an acute angle.
Solution 33
Question 34
If tan 4Ө = cot (Ө + 20°), find the value of Ө if 4Ө is an acute angle.
Solution 34
Question 35
If sec 2Ө = cosec 3Ө, find the value of Ө if it is known that both 2Ө and 3Ө are acute angles.
Solution 35
Question 36
If sin (Ө - 15°) = cos (Ө - 25°), find the value of Ө if (Ө -15°) and (Ө - 25°) are acute angles.
Solution 36
Question 37
Solution 37
Question 38
Solution 38
Question 39
If cosӨ = sin 60° and Ө is an acute angle find the value of 1- 2 sin2Ө
Solution 39
Question 40
If sec Ө = cosec 30° and Ө is an acute angle, find the value of 4 sin2Ө - 2 cos2Ө.
Solution 40
Question 41
Prove the following:
tan Ө tan (90° - Ө) = cot Ө cot (90° - Ө)
Solution 41
Question 42
Prove the following:
sin 58° sec 32° + cos 58° cosec 32° = 2
Solution 42
Question 43
Prove the following:
Solution 43
Question 44
Prove the following:
Solution 44
Question 45
If A + B = 90°, prove that
Solution 45