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Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
Exercise 23(A) Page:291
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.
Solution:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
2. Find the value of :
(i)
tan2 30o + tan2 45o + tan2 60o
(ii)
(iii)
3 sin2 30o + 2 tan2 60o - 5 cos2 45o.
Solution:
(i)
(ii)
(iii)
3 sin2 30o + 2 tan2 60o - 5 cos2 45o
3. 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.
Solution:
(i) LHS=sin 60o cos 30o + cos 60o. sin 30o
=
(ii) LHS=cos 30o. cos 60o - sin 30o. sin 60o
= =RHS
(iii) LHS= cosec2 45o - cot2 45o
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
= =RHS
(iv) LHS= cos2 30o - sin2 30o
= =RHS
(v) LHS=
= =RHS
(vi) LHS=
= =RHS
4. Prove that:
(i) sin (2 30o) =
(ii) cos (2 30o) =
(iii) tan (2 30o) =
Solution:
(i)
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(ii)
(iii)
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
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
Solution:
AB = BC = x
(i)
(ii)
(iii)
6. Prove that:
(i) sin 60o = 2 sin 30o cos 30o.
(ii) 4 (sin4 30o + cos4 60o)-3 (cos2 45o - sin2 90o) = 2
Solution:
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.
Solution:
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(i)
Since angle x is acute, 0<x
(ii)
(iii)
(iv)
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 + cos2 .
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
Solution:
(i)
if x and y are acute angles,
is false.
(ii)
Sec . Cot = cosec is true
(iii)
9. State for any acute angle π whether:
(i) sin π½ increases or decreases as π½ increases:
(ii) cos π½ increases or decreases as π½ increases.
(iii) tan π½ increases or decreases as π½ decreases.
Solution:
(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.
10. If βπ = 1.732, find (correct to two decimal place) the value of each of the following:
(i) sin 60o
(ii)
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
Solution:
(i)
(ii)
11. Evaluate:
(i)
, when A = 15o.
(ii)
; when B = 20o.
Solution:
(i) Given that A=150
(ii) Given that B=200
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
Exercise 23(B) Page:293
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) =
Solution:
A = 60o and B = 30o
(i) According to the question,
(ii)
Hence Proved
(iii) According to the question,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
Hence Proved
(iv) According to the question,
Hence Proved
2. If A =30o, then prove that:
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(i) sin 2A = 2sin A cos A = πππππ¨
π+πππππ¨
(ii) cos 2A = cos2A - sin2A= πβπππππ¨
π+πππππ¨
(iii) 2 cos2 A - 1 = 1 - 2 sin2A
(iv) sin 3A = 3 sin A - 4 sin3A.
Solution:
Given A=300
(i) According to the question,
Hence Proved
(ii) According to the question,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
Hence Proved
(iii) According to the question,
Hence Proved
(iv) According to the question,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
Hence Proved
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
Solution:
A = B = 45o
(i) According to the question,
Hence Proved
(ii) According to the question,
Hence Proved
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) πβπππ ππ¨
πππ ππ¨= ππππ¨
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(v) π+πππππ¨+πππππ¨
ππππ¨+πππ π¨ = ππππ π¨
(vi) 4 cos A cos (60o - A). cos (60o + A) = cos 3A
(vii) πππ 3π΄βπππ 3π΄
πππ π΄+
π ππ3π΄βπ ππ3π΄
π πππ΄= 3
Solution:
A = 30o
(i) According to the question,
Hence Proved.
(ii) According to the question,
We have,
Hence Proved
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(iii) According to the question,
We have,
Hence Proved
(iv) According to the question,
We have,
Hence Proved
(v) According to the question,
We have,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
Hence Proved
(vi) According to the question,
We have,
Hence Proved
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(vii) According to the question,
We have,
Hence Proved
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
Exercise 23(C) Page:297
1. Solve the following equations for A, if :
(i) 2 sin A = 1
(ii) 2 cos 2 A = 1
(iii) sin 3 A = βπ
π
(iv) sec 2 A = 2
(v) βπ tan A = 1
(vi) tan 3 A = 1
(vii) 2 sin 3 A = 1
(viii) βπ cot 2 A = 1
Solution:
(viii) According to the question,
We have,
(ix) According to the question,
We have,
(x) According to the question,
We have,
(xi) According to the question,
We have,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(xii) According to the question,
We have,
(xiii) According to the question,
We have,
(xiv) According to the question,
We have,
(xv) According to the question,
We have,
2. Calculate the value of A, if :
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(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
Solution:
(i) According to the question,
(ii) According to the question,
(iii) According to the question,
(iv) According to the question,
(v) According to the question,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
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.
Solution:
(i) According to the question,
(ii) According to the question,
(iii) According to the question,
4. If 4 cos2 xo - 1 = 0 and 0 xo 90o, find:
(i) xo
(ii) sin2 xo + cos2 xo
(iii) π
πππππΒ°β πππππΒ°
Solution:
(i)
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(ii)
(iii)
5. If 4 sin2 π½ - 1= 0 and angle π½ is less than 90o, find the value of π½ and hence the value of
cos2 π½ + tan2 π½.
Solution:
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
6. If sin 3A = 1 and 0 < A < 90o, find:
(i) sin A
(ii) cos 2A
(iii) π‘ππ2π΄ β1
πππ 2π΄
Solution:
According to the question,
Since sin 3A = 1,
We have,
(i)
(ii)
(iii)
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
7. If 2 cos 2A = βπ and A is acute, find:
(i) A
(ii) sin 3A
(iii) sin2 (75o - A) + cos2 (45o +A)
Solution:
(i) According to the question,
(ii) According to the question,
(iii) According to the question,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
8.
(i) If sin x + cos y = 1 and x = 30o, find the value of y.
(ii) If 3 tan A - 5 cos B= βπ and B = 90o, find the value of A.
Solution:
(i) According to the question,
x = 30o
(ii) According to the question,
B = 90o
9. From the given figure, find:
(i) cos xo
(ii) xo
(iii) π
πππππΒ°β
π
πππππΒ°
(iv) Use tan xo, to find the value of y.
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
Solution:
(i) According to the question,
(ii) According to the question,
(iii) According to the question,
(iv) According to the question,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
10. Use the given figure to find:
(i) tan π½o
(ii) π½o
(iii) sin2 π½o - cos2 π½o
(iv) Use sin π½o to find the value of x.
Solution:
(i) According to the question,
(ii) According to the question,
(iii) According to the question,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(iv) According to the 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
Solution:
(i) According to the question,
(ii) According to the question,
(iii) According to the question,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(iv) According to the question,
12. Solve for x:
(i) 2 cos 3x - 1 = 0
(ii) Cos π
πβ π = 0
(iii) sin (x + 10o) = Β½
(iv) cos (2x - 30o) = 0
(v) 2 cos (3x - 15o) = 1
(vi) tan2 (x - 5o) = 3
(vii) 3 tan2 (2x - 20o) = 1
(viii) Cos (π
π+ ππ) =
βπ
π
(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
Solution:
(i) According to the question,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(ii) According to the question,
(iii) According to the question,
(iv) According to the question,
(v) According to the question,
(vi) According to the question,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(vii) According to the question,
(viii) According to the question,
(ix) According to the question,
(x) According to the question,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(xi) According to the question,
(xii) According to the 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
Solution:
(i) According to the question,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(ii) According to the question,
(iii) According to the question,
(iv) According to the question,
14. In triangle ABC, angle B = 90o, AB = y units, BC = βπ 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.
Solution:
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles
(i)
From triangle ABC
(ii) According to the question,
(iii) According to the question,
(iv) According to the question,
15. If 2 cos (A + B) = 2 sin (A - B) = 1; find the values of A and B.
Solution:
According to the question,
Adding equation (1) and equation (2),
We get,
Concise Selina Solutions for Class 9 Maths Chapter 23-
Trigonometrical Ratios of Standard Angles