# FRANK Solutions for Class 9 Maths Chapter 27 - Trigonometrical Ratios of Standard Angles

Complete your revision more effectively with Frank Solutions for ICSE Class 9 Mathematics Chapter 27 Trigonometrical ratios of standard angles. Check out the answers by experts to understand how they have provided proofs or presented an evaluation of trigonometric ratios given in the Frank textbook questions. The expert solutions on TopperLearning can be accessed 24x7 online.

In addition, the Frank solutions for Chapter 27 also provide elaborate answers on finding the angle or side of a triangle using trigonometry. To further revise this ICSE Class 9 Maths chapter or chapters from other syllabus, take a look at our Selina textbook solutions, doubts and solutions, solved question papers, and other useful resources.

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

Question 1
Solution 1
Question 2
Solution 2
Question 3(a)

Prove that :

sin 60° .cos 30° - sin 60°. sin 30° =

Solution 3(a)

Question 3(b)

Prove that :

cos 60° . cos 30° - sin 60° . sin 30° = 0

Solution 3(b)

Question 3(c)

Prove that :

sec245° - tan245° = 1

Solution 3(c)

Question 3(d)

Prove that :

Solution 3(d)

Question 4(a)

Find the value of 'A', if

2 cos A = 1

Solution 4(a)

Question 4(b)

Find the value of 'A', if

2 sin 2A = 1

Solution 4(b)

Question 4(c)

Find the value of 'A', if

Solution 4(c)

Question 4(d)

Find the value of 'A', if

2 cos 3A = 1

Solution 4(d)

Question 4(e)

Find the value of 'A', if

Solution 4(e)

Question 4(f)

Find the value of 'A', if

cot 3A = 1

Solution 4(f)

Question 5(a)

Find the value of 'A', if

(1 - cosec A)(2 - sec A) = 0

Solution 5(a)

Question 5(b)

Find the value of 'A', if

(2 - cosec 2A) cos 3A = 0

Solution 5(b)

Question 6

If sin α + cosβ = 1 and α= 90°, find the value of 'β'.

Solution 6

Question 7(a)

Solve for 'Ө':

Solution 7(a)

Question 7(b)

Solve for 'Ө':

cot2(Ө - 5)° = 3

Solution 7(b)

Question 7(c)

Solve for 'Ө':

Solution 7(c)

Question 8

Solution 8

Question 9

If sin Ө = cosӨ and 0° < Ө<90°, find the value of 'Ө'.

Solution 9

Question 10

If tan Ө= cot Ө and 0°≤Ө≤ 90°, find the value of 'Ө'.

Solution 10

Question 11

Solution 11

Question 12(a)

If Ө = 30°, verify that:

Solution 12(a)

Question 12(b)

If Ө = 30°, verify that:

Solution 12(b)

Question 12(c)

If A = 30°, verify that:

Solution 12(c)

Question 12(d)

If Ө = 30°, verify that:

sin 3Ө = 4sin Ө.sin(60° - Ө) sin(60° + Ө)

Solution 12(d)

Question 12(e)

If Ө = 30°, verify that:

1 - sin 2Ө= (sin Ө - cosӨ)2

Solution 12(e)

Question 13(a)

Evaluate the following:

Solution 13(a)

Question 13(b)

Evaluate the following:

Solution 13(b)

Question 14

If Ө = 15°, find the value of:

cos 3Ө - sin 6Ө + 3sin (5Ө + 15°) - 2 tan23Ө

Solution 14

Question 15

Solution 15

Question 16(a)

If A = 30° and B = 60°, verify that:

sin (A + B) = sin A cos B + cos A sin B

Solution 16(a)

Question 16(b)

If A = 30° and B = 60°, verify that:

cos (A + B) = cos A cos B - sin A sin B

Solution 16(b)

Question 16(c)

If A = 30° and B = 60°, verify that:

Solution 16(c)

Question 16(d)

If A = 30° and B = 60°, verify that:

Solution 16(d)

Question 17(a)

If A = B = 45°, verify that

sin (A - B) = sin A .cos B - cos A.sin B

Solution 17(a)

Question 17(b)

If A = B = 45°, verify that

cos (A - B) = cosA.cos B + sin A.sin B

Solution 17(b)

Question 18

Solution 18

Question 19(a)

If Ө < 90°, find the value of:

sin2Ө + cos2Ө

Solution 19(a)

Question 19(b)

If Ө < 90°, find the value of:

Solution 19(b)

Question 20(a)

Ifsec 2Ө = 2 and Ө< 90°, find the value of

Ө

Solution 20(a)

Question 20(b)

Ifsec 2Ө = 2 and Ө< 90°, find the value of

cos 3Ө

Solution 20(b)

Question 20(c)

Ifsec 2Ө = 2 and Ө< 90°, find the value of

cos2 (30° + Ө) + sin2 (45° - Ө)

Solution 20(c)

Question 21

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 21

Question 22

Find the value of:

If 3 tan2Ө - 1 = 0, find the value

a. cos 2Ө

b. sin 2Ө

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

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

Question 14

Solution 14

Question 15

Solution 15

2

Question 16

Solution 16

Question 1(a)

Find the value of 'x' in each of the following:

Solution 1(a)

Question 1(b)

Find the value of 'x' in each of the following:

Solution 1(b)

Question 1(c)

Find the value of 'x' in each of the following:

Solution 1(c)

Question 1(d)

Find the value of 'x' in each of the following:

Solution 1(d)

Question 2

Given: ABC = 60°, DBC = 45° and BC = 24 cm.

Solution 2

Question 3

Find lengths of diagonals AC and BD. Given AB = 24 cm and BAD = 60°.

Solution 3

Question 4

In a trapezium ABCD, as shown, ABDC, AD = DC = BC = 24 cm and A = 30°. Find:

length of AB

Solution 4

Construction: Draw DP AB and CM AB

Question 5

Find the length of EC.

Solution 5

Question 6

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 6

Question 7

In the given figure, B = 60°, C = 30°, AB = 8 cm and BC = 24 cm. Find:

a. BE

b. AC

Solution 7

Question 8

Find:

a. BC

c. AC

Solution 8

Question 9(a)

Find the value 'x', if:

Solution 9(a)

Question 9(b)

Find the value 'x', if:

Solution 9(b)

Question 9(c)

Find the value 'x', if:

Solution 9(c)

Question 9(d)

Find the value 'x', if:

Solution 9(d)

Question 9(e)

Find the value 'x', if:

Solution 9(e)

Question 9(f)

Find the value 'x', if:

Solution 9(f)

Question 10(a)

Solution 10(a)

Construction: Draw BX AE

Then, BD = EX = 14 cm and BX = ED

AX = AE - EX = 16 - 14 = 2

Question 10(b)

Solution 10(b)

Question 11

Solution 11

Question 12(a)

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

Solution 12(a)

Question 12(b)

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

Solution 12(b)

Question 13

Solution 13

Question 17

In right-angled triangle ABC; B = 90°. Find the magnitude of angle A, if:

Solution 17

Consider the following figure,

Question 18

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 18

Question 19

The perimeter of a rhombus is 100 cm and obtuse angle of it is 120°. Find the lengths of its diagonals.

Solution 19

Consider the following figure,

Question 20

In the given figure; B = 90°, ADB = 30°, ACB = 45° and AB = 24 m.

Find the length of CD.

Solution 20

Question 21

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 21

Draw QM RS.

Clearly, RQM = 30°

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

Question 1(a)

Evaluate the following:

Solution 1(a)

Question 1(b)

Evaluate the following:

Solution 1(b)

Question 1(c)

Evaluate the following:

Solution 1(c)

Question 1(d)

Evaluate the following:

Solution 1(d)

Question 1(e)

Evaluate the following:

Solution 1(e)

Question 1(f)

Evaluate the following:

Solution 1(f)

Question 2(a)

Evaluate the following:

sin 31° - cos 59°

Solution 2(a)

Question 2(b)

Evaluate the following:

cot 27° - tan 63°

Solution 2(b)

Question 2(c)

Evaluate the following:

cosec 54° - sec 36°

Solution 2(c)

Question 2(d)

Evaluate the following:

sin 28° sec 62° + tan 49° tan 41°

Solution 2(d)

Question 2(e)

Evaluate the following:

sec 16° tan 28° - cot 62° cosec 74°

Solution 2(e)

Question 2(f)

Evaluate the following:

sin 22° cos 44° - sin 46° cos 68°

Solution 2(f)

Question 3(a)

Evaluate the following:

Solution 3(a)

Question 3(b)

Evaluate the following:

Solution 3(b)

Question 3(c)

Evaluate the following:

Solution 3(c)

Question 3(d)

Evaluate the following:

Solution 3(d)

Question 4(a)

Express each of the following in terms of trigonometric ratios of angles between 0° and 45°:

sin 65° + cot 59°

Solution 4(a)

Question 4(b)

Express each of the following in terms of trigonometric ratios of angles between 0° and 45°:

cos 72° - cos 88°

Solution 4(b)

Question 4(c)

Express each of the following in terms of trigonometric ratios of angles between 0° and 45°:

cosec 64° + sec 70°

Solution 4(c)

Question 4(d)

Express each of the following in terms of trigonometric ratios of angles between 0° and 45°:

tan 77° - cot 63° + sin 57°

Solution 4(d)

Question 4(e)

Express each of the following in terms of trigonometric ratios of angles between 0° and 45°:

sin 53° + sec 66° - sin 50°

Solution 4(e)

Question 4(f)

Express each of the following in terms of trigonometric ratios of angles between 0° and 45°:

cos 84° + cosec 69° - cot 68°

Solution 4(f)

Question 5(a)

Evaluate the following:

sin 35° sin 45° sec 55° sec 45°

Solution 5(a)

Question 5(b)

Evaluate the following:

cot 20° cot 40° cot 45° cot 50° cot 70°

Solution 5(b)

Question 5(c)

Evaluate the following:

cos 39° cos 48° cos 60° cosec 42° cosec 51°

Solution 5(c)

Question 5(d)

Evaluate the following:

sin (35° + Ө) - cos (55° - Ө) - tan (42° + Ө) + cot (48° - Ө)

Solution 5(d)

Question 5(e)

Evaluate the following:

tan (78° + Ө) + cosec (42° + Ө) - cot (12° - Ө) - sec(48° - Ө)

Solution 5(e)

Question 5(f)

Evaluate the following:

Solution 5(f)

Question 5(g)

Evaluate the following:

Solution 5(g)

Question 5(h)

Evaluate the following:

Solution 5(h)

Question 5(i)

Evaluate the following:

Solution 5(i)

Question 5(j)

Evaluate the following:

Solution 5(j)

Question 6

If cos 3Ө = sin (Ө - 34°), find the value of Ө if 3Ө is an acute angle.

Solution 6

Question 7

If tan 4Ө = cot (Ө + 20°), find the value of Ө if 4Ө is an acute angle.

Solution 7

Question 8

If sec 2Ө = cosec 3Ө, find the value of Ө if it is known that both 2Ө and 3Ө are acute angles.

Solution 8

Question 9

If sin (Ө - 15°) = cos (Ө - 25°), find the value of Ө if (Ө -15°) and (Ө - 25°) are acute angles.

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

If cosӨ = sin 60° and Ө is an acute angle find the value of 1- 2 sin2Ө

Solution 12

Question 13

If sec Ө = cosec 30° and Ө is an acute angle, find the value of 4 sin2Ө - 2 cos2Ө.

Solution 13

Question 14(a)

Prove the following:

tan Ө tan (90° - Ө) = cot Ө cot (90° - Ө)

Solution 14(a)

Question 14(b)

Prove the following:

sin 58° sec 32° + cos 58° cosec 32° = 2

Solution 14(b)

Question 14(c)

Prove the following:

Solution 14(c)

Question 14(d)

Prove the following:

Solution 14(d)

Question 15

If A + B = 90°, prove that

Solution 15

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