Chapter 22 : Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and their Reciprocals] - Selina Solutions for Class 9 Maths ICSE

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Chapter 22 - Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and their Reciprocals] Excercise Ex. 22(A)

Question 1

From the following figure, find the values of :

(i) sin A

(ii) cos A

(iii) cot A

(iv) sec C

(v) cosec C

(vi) tan C.

Solution 1

Given angle

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Question 2

Form the following figure, find the values of :

(i) cos B

(ii) tan C

(iii) sin2B + cos2B

(iv) sin B. cos C + cos B. sin C

Solution 2

Given angle

(i)

(ii)

(iii)

(iv)

Question 3

From the following figure, find the values of :

(i) cos A (ii) cosec A

(iii) tan2A - sec2A (iv) sin C

(v) sec C (vi) cot2 C -

Solution 3

Consider the diagram as

 

Given angle and

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Question 4

From the following figure, find the values of :

(i) sin B (ii) tan C

(iii) sec2 B - tan2B (iv) sin2C + cos2C

Solution 4

Given angle and

(i)

(ii)

(iii)

(iv)

Question 5

Given: sin A = , find :

(i) tan A (ii) cos A

Solution 5

Consider the diagram below:

Therefore if length of , length of

Since

Now

(i)

(ii)

Question 6

From the following figure, find the values of :

(i) sin A

(ii) sec A

(iii) cos2 A + sin2A

Solution 6

Given angle in the figure

Now

(i)

(ii)

(iii)

Question 7

Given: cos A =

Evaluate: (i) (ii)

Solution 7

Consider the diagram below:

Therefore if length of , length of

Since

Now

(i)

(ii)

Question 8

Given: sec A = , evaluate : sin A -

Solution 8

Consider the diagram below:

Therefore if length of , length of

Since

Now

Therefore

Question 9

Given: tan A = , find :

Solution 9

Consider the diagram below:


Therefore if length of A B equals 3 x, length of B C equals 4 x

Since


Now


Therefore

Question 10

Given: 4 cot A = 3 find;

(i) sin A

(ii) sec A

(iii) cosec2 A - cot2A.

Solution 10

Consider the diagram below:

 


Therefore if length of AB = 3x, length of BC = 4x

Since

(i)

(ii)

(iii)

Question 11

Given: cos A = 0.6; find all other trigonometrical ratios for angle A.

Solution 11

Consider the diagram below:


Therefore if length of AB = 3x, length of AC = 5x

Since


Now all other trigonometric ratios are

Question 12

In a right-angled triangle, it is given that A is an acute angle and tan A =.

find the value of :

(i) cos A(ii) sin A(iii)

Solution 12

Consider the diagram below:


Therefore if length of AB = 12x, length of BC = 5x

Since


(i)

(ii)

(iii)

Question 13

Given: sin

Find cos + sin in terms of p and q.

Solution 13

Consider the diagram below:


Therefore if length of perpendicular = px, length of hypotenuse = qx


Since


Now


Therefore

Question 14

If cos A = and sin B = , find the value of :

Are angles A and B from the same triangle? Explain.

Solution 14

Consider the diagram below:


Therefore if length of AB = x, length of AC = 2x

Since


Consider the diagram below:

 


Therefore if length of AC = x, length of B C equals square root of 2 x 

Since


Now


Therefore

Question 15

If 5 cot = 12, find the value of : Cosec + sec

Solution 15

Consider the diagram below:


Therefore if length of base = 12x, length of perpendicular = 5x


Since


Now


Therefore

Question 16

If tan x = , find the value of : 4 sin2x - 3 cos2x + 2

Solution 16

Consider the diagram below:


Therefore if length of base = 3x, length of perpendicular = 4x


Since


Now


Therefore

Question 17

Ifcosec = , find the value of:

(i) 2 - sin2 - cos2

(ii)

Solution 17

Consider the diagram below:

 


Therefore if length of hypotenuse equals square root of 5 x, length of perpendicular = x

Since


Now

(i)

 

(ii)

Question 18

If sec A = , find the value of :

Solution 18

Consider the diagram below:


Therefore if length of AB = x, length of A C equals square root of 2 x 

Since


Now


Therefore

Question 19

If cot = 1; find the value of: 5 tan2 + 2 sin2 - 3

Solution 19

Consider the diagram below:


Therefore if length of base = x, length of perpendicular = x


Since


Now


Therefore

Question 20

In the following figure:

AD BC, AC = 26 CD = 10, BC = 42,

DAC = x and B = y.

Find the value of :

(i) cot x

(ii)

(iii)

Solution 20

Given angle and in the figure


Again


Now

(i)

(ii)


Therefore


(iii)


Therefore

Chapter 22 - Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and their Reciprocals] Excercise Ex. 22(B)

Question 1

From the following figure, find:

(i) y (ii) sin xo

(iii) (sec xo - tan xo) (sec xo + tan xo)

Solution 1

Consider the given figure

 

(i)

Since the triangle is a right angled triangle, so using Pythagorean Theorem

(ii)

(iii)

Therefore

Question 2

Use the given figure to find:

(i) sin xo (ii) cos yo

(iii) 3 tan xo - 2 sin yo + 4 cos yo.

Solution 2

Consider the given figure

Since the triangle is a right angled triangle, so using Pythagorean Theorem

Also

(i)

(ii)

(iii)

Therefore

Question 3

In the diagram, given below, triangle ABC is right-angled at B and BD is perpendicular to AC. Find:

(i) cos DBC (ii) cot DBA

Solution 3

Consider the given figure

Since the triangle is a right angled triangle, so using Pythagorean Theorem

In and , the is common to both the triangles, so therefore .

Therefore and are similar triangles according to AAA Rule

So

(i)

(ii)

Question 4

In the given figure, triangle ABC is right-angled at B. D is the foot of the perpendicular from B to AC. Given that BC = 3 cm and AB = 4 cm. find:

(i) tan DBC

(ii) sin DBA

Solution 4

Consider the given figure

 

Since the triangle is a right angled triangle, so using Pythagorean Theorem

In and , the is common to both the triangles, so therefore.

Therefore and are similar triangles according to AAA Rule

So

Now using Pythagorean Theorem

Therefore


(i) 

 

 

 

(ii)

Question 5

In triangle ABC, AB = AC = 15 cm and BC = 18 cm, find cos ABC.

Solution 5

Consider the figure below

In the isosceles , and the perpendicular drawn from angle to the side divides the side into two equal parts

Question 6

In the figure given below, ABC is an isosceles triangle with BC = 8 cm and AB = AC = 5 cm. Find:

(i) sin B (ii) tan C

(iii) sin2 B + cos2B (iv) tan C - cot B

Solution 6

Consider the figure below

In the isosceles , and the perpendicular drawn from angle to the side divides the side into two equal parts

Since

(i)

(ii)

(iii)

Therefore

(iv)

Therefore

Question 7

In triangle ABC; ABC = 90o, CAB = xo, tan xo = and BC = 15 cm. Find the measures of AB and AC.

Solution 7

Consider the figure


Therefore if length of base = 4x, length of perpendicular = 3x


Since


Now


Therefore


And

Question 8

Using the measurements given in the following figure:

(i) Find the value of sin and tan.

(ii) Write an expression for AD in terms of

Solution 8

Consider the figure

 

A perpendicular is drawn from D to the side AB at point E which makes BCDE is a rectangle.

Now in right angled triangle BCD using Pythagorean Theorem

Since BCDE is rectangle so ED 12 cm, EB = 5 and AE = 14 - 5 = 9 

(i)

(ii)

Or

Question 9

In the given figure;

BC = 15 cm and sin B =.

(i) Calculate the measure of AB and AC.

(ii) Now, if tan ADC = 1; calculate the measures of CD and AD.

Also, show that: tan2B -

Solution 9

Given



Therefore if length of perpendicular = 4x, length of hypotenuse = 5x


Since

Now

(i)

And

(ii)

Given


Therefore if length of perpendicular = x, length of hypotenuse = x


Since

Now

So

And


Now


So

Question 10

If sin A + cosec A = 2;

Find the value of sin2 A + cosec2 A.

Solution 10

Squaring both sides

Question 11

If tan A + cot A = 5;

Find the value of tan2 A + cot2 A.

Solution 11

Squaring both sides

Question 12

Given: 4 sin = 3 cos ; find the value of:

(i) sin (ii) cos

(iii) cot2 - cosec2.

(iv) 4 cos2- 3 sin2+ 2

Solution 12

Consider the diagram below:


Therefore if length of BC = 3x, length of AB = 4x


Since

(i)

(ii)

(iii)

Therefore

(iv)

Question 13

Given : 17 cos = 15;

Find the value of: tan + 2 sec.

Solution 13

Consider the diagram below:


Therefore if length of AB = 15x, length of AC = 17x


Since


Now


Therefore

Question 14

Given : 5 cos A - 12 sin A = 0; evaluate  :

.

Solution 14

Now

Question 15

In the given figure; C = 90o and D is mid-point of AC. Find

(i) (ii)

Solution 15

Since is mid-point of so

(i)

(ii)

Question 16

If 3 cos A = 4 sin A, find the value of :

(i) cos A(ii) 3 - cot2 A + cosec2A.

Solution 16

Consider the diagram below:


Therefore if length of AB = 4x, length of BC = 3x


Since


(i)


(ii)


Therefore

Question 17

In triangle ABC, B = 90o and tan A = 0.75. If AC = 30 cm, find the lengths of AB and BC.

Solution 17

Consider the figure


Therefore if length of base = 4x, length of perpendicular = 3x


Since


Now


Therefore


And

Question 18

In rhombus ABCD, diagonals AC and BD intersect each other at point O.

If cosine of angle CAB is 0.6 and OB = 8 cm, find the lengths of the the side and the diagonals of the rhombus.

Solution 18

Consider the figure

The diagonals of a rhombus bisects each other perpendicularly


Therefore if length of base = 3x, length of hypotenuse = 5x


Since


Now


Therefore

And


Since the sides of a rhombus are equal so the length of the side of the rhombus


The diagonals are

Question 19

In triangle ABC, AB = AC = 15 cm and BC = 18 cm. Find:

(i) cos B (ii) sin C

(iii) tan2 B - sec2 B + 2

Solution 19

Consider the figure below

In the isosceles , the perpendicular drawn from angle to the side divides the side into two equal parts

Since

(i)

(ii)

(iii)

Therefore

Question 20

In triangle ABC, AD is perpendicular to BC. sin B = 0.8, BD = 9 cm and tan C = 1. Find the length of AB, AD, AC and DC.

Solution 20

Consider the figure below


Therefore if length of perpendicular = 4x, length of hypotenuse = 5x


Since


Now


Therefore


And

Again


Therefore if length of perpendicular = x, length of base = x


Since


Now


Therefore


And

Question 21

Given q tan A = p, find the value of :

.

Solution 21

Now

Question 22

If sin A = cos A, find the value of 2 tan2A - 2 sec2 A + 5.

Solution 22

Consider the figure


Therefore if length of perpendicular = x, length of base = x


Since


Now


Therefore

Question 23

In rectangle ABCD, diagonal BD = 26 cm and cotangent of angle ABD = 1.5. Find the area and the perimeter of the rectangle ABCD.

Solution 23

Consider the diagram


Therefore if length of base = 3x, length of perpendicular = 2x


Since


Now


Therefore


Now

Question 24

If 2 sin x = , evaluate.

(i) 4 sin3 x - 3 sin x.

(ii) 3 cos x - 4 cos3 x.

Solution 24

Consider the figure

Therefore if length of , length of

Since

Now

(i)

(ii)

Question 25

If sin A = and cos B = , find the value of : .

Solution 25

Consider the diagram below:

Therefore if length of , length of

Since

Consider the diagram below:

Therefore if length of , length of

Since

Now

Therefore

Question 26

Use the informations given in the following figure to evaluate:

Solution 26

Consider the given diagram as

Using Pythagorean Theorem

Now

Again using Pythagorean Theorem

Now

Therefore

Question 27

If sec A = , find: .

Solution 27

Consider the figure

Therefore if length of , length of

Since

Now

Therefore

Question 28

If 5 cos = 3, evaluate : .

Solution 28

Now

Question 29

If cosec A + sin A = 5, find the value of cosec2A + sin2A.

Solution 29

Squaring both sides

Question 30

If 5 cos = 6 sin ; evaluate:

(i) tan (ii)

Solution 30

Now

(i)

(ii)

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