# Trigonometric Ratios Of Standard Angles Class-9 Concise ICSE Maths

Trigonometric Ratios Of Standard Angles Class-9 Concise Selina ICSE Mathematics Solutions Chapter-23 . We provide step by step Solutions of Exercise / lesson-23 Trigonometric Ratios Of Standard Angles for ICSE Class-9 Concise Selina Mathematics by R K Bansal.

Our Solutions contain all type Questions with Exe-23 A and Exe-23 B,  Exe-23 C  to develop skill and confidence. Visit official Website CISCE for detail information about ICSE Board Class-9 Mathematics .

## Trigonometric Ratios Of Standard Angles Class-9 Concise Selina ICSE Mathematics Solutions Chapter-23

–: Select Topics :–

Exe-23 A,

Exe-23 B,

Exe-23 C,

### Exercise 23(A), Trigonometric Ratios Of Standard Angles Class-9 Concise Selina ICSE Mathematics Solutions

Question 1

find the value of:

(i) sin 30o cos 30o

(ii) tan 30o tan 60o

(iii) cos2 60o + sin2 30o

(iv) cosec2 60o – tan2 30o

(v) sin2 30o + cos2 30o + cot2 45o

(vi) cos2 60o + sec2 30o + tan2 45o.

Question 2

find the value of :

(i) tan2 30o + tan2 45o + tan2 60o

(ii) …………..

(iii) 3 sin2 30o + 2 tan2 60o – 5 cos2 45o.

Solution 2:

Prove that:

(i) sin 60o cos 30o + cos 60o. sin 30o = 1

(ii) cos 30o. cos 60o – sin 30o. sin 60o = 0

(iii) cosec2 45o – cot2 45o = 1

(iv) cos2 30o – sin2 30o = cos 60o.

(v) ………………

(vi) 3 cosec2 60o – 2 cot2 30o + sec2 45o = 0.

Question 4

Without using table prove that:

(i) sin (2x 30o) = …………

(ii) cos (2x 30o) = ………….

(iii) tan (2x 30o) = …………..

Question 5

ABC is an isosceles right-angled triangle. Assuming of AB = BC = x, find the value of each of the following trigonometric ratios:

(i) sin 45o

(ii) cos 45o

(iii) tan 45o

…………………

Question 6

Prove that:

(i) sin 60o = 2 sin 30o cos 30o.

(ii) 4 (sin4 30o + cos4 60o)

-3 (cos2 45o – sin2 90o) = 2

Question 7

(i) If sin x = cos x and x is acute, state the value of x.

(ii) If sec A = cosec A and 0o A 90o, state the value of A.

(iii) If tan = cot and 0o 90o, state the value of.

(iv) If sin x = cos y; write the relation between x and y, if both the angles x and y are acute.

Question 8

(i) If sin x = cos y, then x + y = 45o ; write true of false.

(ii) sec. Cot = cosec; write true or false.

(iii) For any angle , state the value of :

Sin2  θ+ cos2

Question 9

State for any acute angle  whether:

(i) sin θ increases or decreases as θ increases:

(ii) cos θ increases or decreases as θ increases.

(iii) tan θ increases or decreases as θ decreases.

(i) For acute angles, remember what sine means: opposite over hypotenuse. If we increase the angle, then the opposite side gets larger. That means “opposite/hypotenuse” gets larger or increases.
(ii) For acute angles, remember what cosine means: base over hypotenuse. If we increase the angle, then the hypotenuse side gets larger. That means “base/hypotenuse” gets smaller or decreases.
(iii) For acute angles, remember what tangent means: opposite over base. If we decrease the angle, then the opposite side gets smaller. That means “opposite /base” gets decreases.

Question 10

If √3 = 1.732, find (correct to two decimal place) the value of each of the following:

(i) sin 60o (ii)  2/ tan 30

Question 11

Evaluate:

(i) , ………….when A = 15o.

(ii)  ;……………… when B = 20o.

### Trigonometric Ratios Of Standard Angles Class-9 Concise Selina ICSE Mathematics Solutions  Exercise 23(B)

Question 1

Given A = 60o and B = 30o, prove that:

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

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

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

(iv) tan (A – B) = …………..

Question 2

If A =30o, then prove that:

(i) sin 2A = 2sin A cos A = …………..

(ii) cos 2A = cos2A – sin2A

= ……………….

(iii) 2 cos2 A – 1 = 1 – 2 sin2A

(iv) sin 3A = 3 sin A – 4 sin3A.

Question 3

If A = B = 45o, show that:

(i) sin (A – B) = sin A cos B – cos A sin B

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

Question 4

If A = 30o; show that:

(i) sin 3 A

= 4 sin A sin (60o – A) sin (60o + A)

(ii) (sin A – cos A)2 = 1 – sin 2A

(iii) cos 2A = cos4 A – sin4 A

(iv) ………….

(v) ……………= 2 cos A.

(vi) 4 cos A cos (60o – A). cos (60o + A)

= cos 3A

(vii) …………… =3

### Trigonometric Ratios Of Standard Angles Exe-23 (C) for Class-9 Concise Selina ICSE Mathematics Solutions

Question 1

Solve the following equations for A, if :

(i) 2 sin A = 1 (ii) 2 cos 2 A = 1

(iii) sin 3 A = √3/2  (iv) sec 2 A = 2

(v) √3 tan A = 1 (vi) tan 3 A = 1

(vii) 2 sin 3 A = 1 (viii)  √3 cot 2 A = 1

Question 2

Calculate the value of A, if :

(i) (sin A – 1) (2 cos A – 1) = 0

(ii) (tan A – 1) (cosec 3A – 1) = 0

(iii) (sec 2A – 1) (cosec 3A – 1) = 0

(iv) cos 3A. (2 sin 2A – 1) = 0

(v) (cosec 2A – 2) (cot 3A – 1) = 0

Question 3

If 2 sin xo – 1 = 0 and xo is an acute angle; find :

(i) sin xo (ii) xo (iii) cos xo and tan xo.

Question 4

If 4 cos2 xo – 1 = 0 and 0  xo  90o, find:

(i) xo (ii) sin2 xo + cos2 xo

(iii) ……………

Question 5

If 4 sin2  θ- 1= 0 and angle  is less than 90o, find the value of  and hence the value of cos2  θ+ tan2

Question 6

If sin 3A = 1 and 0  A  90o, find:

(i) sin A (ii) cos 2A

(iii) tan2A – 1/cos² A

Question 7

If 2 cos 2A = √3 and A is acute, find:

(i) A (ii) sin 3A

(iii) sin2 (75o – A) + cos2 (45o +A)

Question 8

(i) If sin x + cos y = 1 and x = 30o, find the value of y.

(ii) If 3 tan A – 5 cos B= √3 and B = 90o, find the value of A.

Question 9

From the given figure, find:

(i) cos xo(ii) xo

(iii) ……………..

(iv) Use tan xo, to find the value of y.

Question 10

Use the given figure to find:

(i) tan (ii) (iii) sin2o – cos2o

(iv) Use sin o to find the value of x.

Question 11

Find the magnitude of angle A, if:

(i) 2 sin A cos A – cos A – 2 sin A + 1 = 0

(ii) tan A – 2 cos A tan A + 2 cos A – 1 = 0

(iii) 2 cos2 A – 3 cos A + 1 = 0

(iv) 2 tan 3A cos 3A – tan 3A + 1 = 2 cos 3A

Question 12

Solve for x:

(i) 2 cos 3x – 1 = 0 (ii) cos  = 0

(iii) sin (x + 10o) = 1/2 (iv) cos (2x – 30o) = 0

(v) 2 cos (3x – 15o) = 1 (vi) tan2 (x – 5o) = 3

(vii) 3 tan2 (2x – 20o) = 1

(viii) cos ………….. =√3/2

(ix) sin2 x + sin2 30o = 1

(x) cos2 30o + cos2 x = 1

(xi) cos2 30o + sin2 2x = 1

(xii) sin2 60o + cos2 (3x- 9o) = 1

Question 13

If 4 cos2 x = 3 and x is an acute angle; find the value of :

(i) x (ii) cos2 x + cot2 x

(iii) cos 3x (iv) sin 2x

Question 14

In ABC, B = 90o, AB = y units, BC =  units, AC = 2 units and angle A = xo, find:

(i) sin xo (ii) xo (iii) tan xo

(iv) use cos xo to find the value of y.

Question 15

If 2 cos (A + B) = 2 sin (A – B) = 1; find the values of A and B.

— End of Trigonometric Ratios Of Standard Angles Class-9 Concise Selina Solutions :–

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