Exercise 22(A) Page:279 - Byju's...5. In triangle ABC, AB = AC = 15 cm and BC = 18 cm, find cos ABC....

Concise Selina Solutions for Class 9 Maths Chapter 22-

Trigonometrical Ratios

Exercise 22(A) Page:279

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:

Given angle ABC = 900

AC = 5

(i)

(ii)

(iii)

(iv)

https://byjus.com/?utm_source=pdf-click




(v)

(vi)

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:

Given angle BAC = 900

(i)

(ii)

(iii)





(iv)


(i) cos A

(ii) cosec A

(iii) tan2A - sec2A

(iv) sin C

(v) sec C

(vi) cot2 C - 𝟏

𝑺𝒊𝒏𝟐𝑪

Solution:





Given angle ADB and BDC = 900

(i)

(ii)

(iii)

(iv)

(v)

(vi)






(i) sin B

(ii) tan C

(iii) sec2 B - tan2B

(iv) sin2C + cos2C

Solution:

Given angle ADB=900 and ADC = 900





(i)

(ii)

(iii)

(iv)

5. Given: sin A = 3/5 , find :

(i) tan A

(ii) cos A

Solution:





When length of BC=3x, length of AC = 5x

(i)

(ii)


(i) sin A

(ii) sec A

(iii) cos2 A + sin2A

Solution:





(i)

(ii)

(iii)

7. Given: cos A = 5 / 13

Evaluate:

(i)

(ii)

Solution:

When length of AB = 5x, length of AC = 13x





(i)

(ii)

8. Given: sec A = 29/21, evaluate : sin A – (1/tan A)

Solution:






9. Given: tan A = 4/3, find:

Solution:

When the length of AB = 3x, length of BC = 4x





10. Given: 4 cot A = 3 find;

(i) sin A

(ii) sec A

(iii) cosec2 A - cot2A.

Solution:

When length of AB = 3x, length of BC = 4x

(i)

(ii)

(iii)





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

Solution:


Now all other trigonometric ratios are

12. In a right-angled triangle, it is given that A is an acute angle and tan A = 5/12.

find the value of :

(i) cos A

(ii) sin A





(iii) Solution:

When length of AB = 12x, length of BC = 5x

(i)

(ii)

(iii)

13. Given: sin 𝜽 = p/q

Find cos 𝜽 + sin 𝜽 in terms of p and q.

Solution:





When length of perpendicular = px, length of hypotenuse = qx

14. If cos A = ½ and sin B = 1/√𝟐, find the value of :

.

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

Solution:

When length of AB = x, length of AC = 2x





Therefore if length of AC = x, length of BC = √2𝑥

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

Solution:

When length of base = 12x, length of perpendicular = 5x





16. If tan x = 𝟏𝟏

𝟑 , find the value of : 4 sin2x - 3 cos2x + 2

Solution:






17. If cosec 𝜽 = √𝟓, find the value of:

(i) 2 - sin2 - cos2

(ii)

Solution:

When length of hypotenuse = √5𝑥, length of perpendicular = x

(i)





(ii)

18. If sec A = √𝟐, find the value of :

Solution:

When length of AB = x, length of AC = √2𝑥





19. If cot 𝜽= 1; find the value of: 5 tan2𝜽+ 2 sin2 𝜽- 3

Solution:

When length of base = x, length of perpendicular = x

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:

(i)

(ii)





(iii)





Exercise 22(B) Page:285

1. From the following figure, find:

(i) y

(ii) sin xo

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

Solution:

(i)

Using Pythagorean Theorem

(ii)

(iii)





2. Use the given figure to find:

(i) sin xo

(ii) cos yo

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

Solution:






(i)

(ii)

(iii)

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:






In and , the is common to both the triangles,

.

Therefore and are similar triangles according to AAA Rule

So,

(i)

(ii)

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:


In and , the is common to both the triangles,

.

Therefore and are similar triangles according to AAA Rule






(i)

(ii)

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

Solution:

In the isosceles triangle ABC, AB = AC = 15 cm and BC = 18cm the perpendicular drawn from angle A to

the side BC divides the side BC into two equal parts BD=DC=9cm

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:

Since

(i)

(ii)

(iii)





(iv)

7. In triangle ABC; ABC = 90o, CAB = xo, tan xo = ¾ and BC = 15 cm. Find the measures of AB

and AC.

Solution:






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:

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

From triangle BCD, using Pythagorean Theorem

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

(i)

(ii)





Or

9. In the given figure;

BC = 15 cm and sin B = 4/5

(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 – 1/ cos2B = -1

Solution:

Given

When length of perpendicular = 4x, length of hypotenuse = 5x





(i)

(ii)

Given

When length of perpendicular = x, length of hypotenuse = x





10. If sin A + cosec A = 2;

Find the value of sin2 A + cosec2 A.

Solution:

Squaring both sides

11. If tan A + cot A = 5;

Find the value of tan2 A + cot2 A.

Solution:

Squaring both sides

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

(i) sin 𝜽

(ii) cos 𝜽

(iii) cot2 𝜽 - cosec2 𝜽.

(iv) 4 cos2 𝜽 - 3 sin2 𝜽 + 2

Solution:





When length of BC = 3x, length of AB = 4x

(i)

(ii)

(iii)

(iv)





13. Given : 17 cos 𝜽 = 15;

Find the value of: tan 𝜽 + 2 sec 𝜽.

Solution:


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

.

Solution:





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

(i) (ii)

Solution:

Since is mid-point of so

(i)





(ii)

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

(i) cos A

(ii) 3 - cot2 A + cosec2A.

Solution:

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

Since

(i)





(ii)

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

Solution:






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:

When length of base = 3x, length of hypotenuse = 5x





The sides of a rhombus are equal.

So, the length of the side of the rhombus = 10cm

The diagonals are

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:

In the isosceles triangle ABC, the perpendicular drawn from angle A to the side BC divides the side BC into

two equal parts BD = DC = 9cm

Since

(i)

(ii)

(iii)





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:

When length of perpendicular = 4x, length of hypotenuse = 5x





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

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

.

Solution:





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

Solution:

If length of perpendicular = x, length of base = x





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:

If length of base = 3x, length of perpendicular = 2x





24. If 2 sin x = , evaluate.

(i) 4 sin3 x - 3 sin x.

(ii) 3 cos x - 4 cos3 x.

Solution:





(i)

(ii)

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

Solution:









26. Use the informations given in the following figure to evaluate:

Solution:







Then,

27. If sec A = √𝟐,

find: .

Solution:





Then,

28. If 5 cos 𝜽 = 3,

evaluate : .

Solution:





29. If cosec A + sin A = 5𝟏

𝟓, find the value of cosec2A + sin2A.

Solution:

Squaring both sides

30. If 5 cos 𝜽 = 6 sin 𝜽 ; evaluate:

(i) tan 𝜽





(ii)

Solution:

(i)

(ii)



Date post:	12-Aug-2021
Category:	Documents
Upload:	others
View:	3 times
Download:	0 times

Exercise 22(A) Page:279 - Byju's...5. In triangle ABC, AB = AC = 15 cm and BC = 18 cm, find cos ABC....

Documents