SELINA Solutions for Class 9 Maths Chapter 23 - Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios]
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Chapter 23 - Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios] Exercise Ex. 23(A)
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.
Solution 1
(i)
(ii)
(iii)
(iv)
(v)
(vi)
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
(i) 
(ii) 
(iii) 3 sin2 30o + 2 tan2 60o - 5 cos2 45o
Question 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 3
(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
=
=RHS
(iv) LHS= cos2 30o - sin2 30o
=
=RHS
(v) LHS= 
=
=RHS
(vi) LHS= 
=
=RHS
Question 4
prove that:
(i) sin (2 
30o) = 
(ii) cos (2 30o) = 
(iii) tan (2 30o) = 
Solution 4
(i)
(ii)
(iii)
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
Solution 5
Given that AB = BC = x
(i) 
(ii)
(iii)
Question 6
Prove that:
(i) sin 60o = 2 sin 30o cos 30o.
(ii) 4 (sin4 30o + cos4 60o)
-3 (cos2 45o - sin2 90o) = 2
Solution 6
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.
Solution 7
(i)
The angle, x is acute and hence we have, 0 < x
(ii)
(iii)
(iv)
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 + cos2.
Solution 8
(i)
if x and y are acute angles,
is false.
(ii)
Sec. Cot = cosec is true
(iii)
Question 9
State
for any acute angle 
whether:
(i) sin increases or
decreases as increases:
(ii)
cos increases or
decreases as increases.
(iii) tan increases or
decreases as decreases.
Solution 9
(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
= 1.732, find
(correct to two decimal place) the value of each of the following:
(i) sin 60o (ii) 
Solution 10
(i)
(ii) 
Question 11
Evaluate:
(i) 
, when A = 15o.
(ii) 
; when B = 20o.
Solution 11
(i) Given that A=
(ii)
Given that B=
Chapter 23 - Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios] Exercise Ex. 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) = 
Solution 1
Given A = 60o and B = 30o
(i)
(ii)
(iii)
(iv)
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.
Solution 2
Given A=
(i)
(ii)
(iii)
(iv)
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
Solution 3
Given that A = B = 45o
(i)
(ii)
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) 
Solution 4
Given that A = 30o
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
Chapter 23 - Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios] Exercise Ex. 23(C)
Question 1
Solve the following equations for A, if :
(i) 2 sin A = 1 (ii) 2 cos 2 A = 1
(iii)
sin 3 A = 
(iv) sec 2 A = 2
(v)
tan A =
1 (vi) tan 3 A =
1
(vii)
2 sin 3 A = 1
(viii) cot 2 A = 1
Solution 1
(i)
(ii)
(iii)
(iv)
(V)
(vi)
(vii)
(viii)
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
Solution 2
(i)
(ii)
(iii)
(iv)
(v)
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.
Solution 3
(i)
(ii)
(iii)
Question 4
If 4 cos2 xo - 1 = 0 and 0 
xo 90o, find:
(i) xo (ii) sin2 xo + cos2 xo
(iii) 
Solution 4
(i)
(ii)
(iii)
Question 5
If
4 sin2 
- 1= 0 and
angle is less than
90o, find the value of and hence the
value of cos2 + tan2.
Solution 5
Question 6
If
sin 3A = 1 and 0 
A 90o,
find:
(i) sin A (ii) cos 2A
(iii)
tan2A - 
Solution 6
(i)
(ii)
(iii)
Question 7
If 2 cos 2A = 
and A is acute, find:
(i) A (ii) sin 3A
(iii) sin2 (75o - A) + cos2 (45o +A)
Solution 7
(i)
(ii)
(iii)
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= 
and B = 90o, find the value of A.
Solution 8
(i)
Given that x = 30o
(ii)
Given that B = 90o
Question 9
From the given figure, find:
(i) cos xo(ii) xo
(iii) 
(iv) Use tan xo, to find the value of y.
Solution 9
(i)
(ii)
(iii)
(iv)
Question 10
Use the given figure to find:
(i) tan 
o (ii) o (iii) sin2o - cos2o
(iv) Use sin o to find the value of x.
Solution 10
(i)
(ii)
(iii)
(iv)
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 11
(i)
(ii)
(iii)
(iv)
Question 12
Solve for x:
(i) 2 cos 3x - 1 = 0 (ii) cos 
= 0
(iii) sin (x + 10o) = 
(iv) cos (2x - 30o) = 0
(v) 2 cos (3x - 15o) = 1 (vi) tan2 (x - 5o) = 3
(vii) 3 tan2 (2x - 20o) = 1
(viii) cos 
(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 12
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
(xi)
(xii)
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 13
(i)
(ii)
(iii)
(iv)
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.
Solution 14
(i)
From
(ii)
(iii)
(iv)
Question 15
If 2 cos (A + B) = 2 sin (A - B) = 1; find the values of A and B.
Solution 15
Adding (1) and (2)
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