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Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios]

Chapter 23 : Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios] - Selina Solutions for Class 9 Maths ICSE

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Chapter 23 - Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios] Excercise Ex. 23(A)

Question 1

find the value of:

(i) sin 30^o cos 30^o

(ii) tan 30^o tan 60^o

(iii) cos² 60^o + sin² 30^o

(iv) cosec² 60^o - tan² 30^o

(v) sin² 30^o + cos² 30^o + cot² 45^o

(vi) cos² 60^o + sec² 30^o + tan² 45^o.

Solution 1

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Question 2

find the value of :

(i) tan² 30^o + tan² 45^o + tan² 60^o

(ii)

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

Solution 2

(i)

(ii)

(iii) 3 sin² 30^o + 2 tan² 60^o - 5 cos² 45^o

Question 3

Prove that:

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

(ii) cos 30^o. cos 60^o - sin 30^o. sin 60^o = 0

(iii) cosec² 45^o - cot² 45^o = 1

(iv) cos² 30^o - sin² 30^o = cos 60^o.

(v)

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

Solution 3

(i) LHS=sin 60^o cos 30^o + cos 60^o. sin 30^o

(ii) LHS=cos 30^o. cos 60^o - sin 30^o. sin 60^o

==RHS

(iii) LHS= cosec² 45^o - cot² 45^o

==RHS

(iv) LHS= cos² 30^o - sin² 30^o

==RHS

(v) LHS=

==RHS

(vi) LHS=

==RHS

Question 4

prove that:

(i) sin (2 30^o) =

(ii) cos (2 30^o) =

(iii) tan (2 30^o) =

Solution 4

(i)

$R H S equals fraction numerator 2 space tan space 30 degree over denominator 1 plus tan squared 30 degree end fraction equals fraction numerator 2 cross times begin display style fraction numerator 1 over denominator square root of 3 end fraction end style over denominator 1 plus open parentheses begin display style fraction numerator 1 over denominator square root of 3 end fraction end style close parentheses squared end fraction equals fraction numerator begin display style fraction numerator 2 over denominator square root of 3 end fraction end style over denominator 1 plus begin display style 1 third end style end fraction equals fraction numerator fraction numerator 2 over denominator square root of 3 end fraction over denominator begin display style 4 over 3 end style end fraction equals fraction numerator square root of 3 over denominator 2 end fraction L H S equals sin space left parenthesis 2 cross times 30 degree right parenthesis equals sin space 60 degree equals fraction numerator square root of 3 over denominator 2 end fraction therefore L H S thin space equals R H S$

(ii)

$R H S comma fraction numerator 1 minus tan squared 30 degree over denominator 1 plus tan squared 30 degree end fraction equals fraction numerator begin display style 1 minus 1 third end style over denominator 1 plus begin display style 1 third end style end fraction equals 1 half L H S comma cos space left parenthesis 2 cross times 30 degree right parenthesis equals cos space 60 degree equals 1 half L H S thin space equals R H S$

(iii)

$R H S comma fraction numerator 2 space tan space 30 degree over denominator 1 minus tan squared 30 degree end fraction equals fraction numerator 2 begin display style fraction numerator 1 over denominator square root of 3 end fraction end style over denominator 1 minus begin display style 1 third end style end fraction equals fraction numerator begin display style fraction numerator 2 over denominator square root of 3 end fraction end style over denominator begin display style 2 over 3 end style end fraction equals square root of 3 L H S comma tan space left parenthesis 2 cross times 30 degree right parenthesis equals tan space 60 degree equals square root of 3 L H S equals R H S$

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 45^o

(ii) cos 45^o

(iii) tan 45^o

Solution 5

Given that AB = BC = x

(i)

(ii)

(iii)

Question 6

Prove that:

(i) sin 60^o = 2 sin 30^o cos 30^o.

(ii) 4 (sin⁴ 30^o + cos⁴ 60^o)

-3 (cos² 45^o - sin² 90^o) = 2

Solution 6

$left parenthesis i right parenthesis space L H S equals sin space 60 degree equals fraction numerator square root of 3 over denominator 2 end fraction R H S equals 2 space sin space 60 degree cos space 60 degree equals 2 cross times fraction numerator square root of 3 over denominator 2 end fraction cross times 1 half equals fraction numerator square root of 3 over denominator 2 end fraction L H S equals R H S left parenthesis i i right parenthesis space L H S equals 4 left parenthesis sin to the power of 4 30 degree plus cos to the power of 4 60 degree right parenthesis minus 3 open parentheses cos squared 45 degree minus sin squared 90 degree close parentheses equals 4 open square brackets open parentheses 1 half close parentheses to the power of 4 plus open parentheses 1 half close parentheses to the power of 4 close square brackets minus 3 open square brackets open parentheses fraction numerator 1 over denominator square root of 2 end fraction close parentheses squared plus open parentheses 1 close parentheses to the power of 4 close square brackets equals 4 open square brackets 1 over 16 plus 1 over 16 close square brackets minus 3 open square brackets 1 half minus 1 close square brackets equals fraction numerator 4 cross times 2 over denominator 16 end fraction plus 3 cross times 1 half equals 2 R H S equals 2 L H S equals R H S$

Question 7

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

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

(iii) If tan = cot and 0^o 90^o, 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

$W e space k n o w space t h a t cos squared x plus sin squared x equals 1 rightwards double arrow 2 sin squared x equals 1 space space space space space space open square brackets sin c e space cos x equals sin x close square brackets rightwards double arrow sin x equals fraction numerator 1 over denominator square root of 2 end fraction rightwards double arrow x equals 45 degree$

(ii)

(iii)

(iv)

$sin space straight x equals cos space straight y equals sin space open parentheses 90 degree minus straight y close parentheses If space straight x space and space straight y space are space acute space angles comma straight x equals 90 degree minus straight y rightwards double arrow straight x plus straight y equals 90 degree Hence space straight x space and space straight y space are space complementary space angles$

Question 8

(i) If sin x = cos y, then x + y = 45^o ; 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 60^o (ii)

Solution 10

(i)

(ii)

Question 11

Evaluate:

(i) , when A = 15^o.

(ii) ; when B = 20^o.

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] Excercise Ex. 23(B)

Question 1

Given A = 60^o and B = 30^o, 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 = 60^o and B = 30^o

(i)

(ii)

(iii)

(iv)

Question 2

If A =30^o, 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 = 45^o, 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 = 45^o

(i)

(ii)

Question 4

If A = 30^o; show that:

(i) sin 3 A

= 4 sin A sin (60^o - A) sin (60^o + 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 (60^o - A). cos (60^o + A)

= cos 3A

(vii)

Solution 4

Given that A = 30^o

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

Chapter 23 - Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios] Excercise 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 x^o - 1 = 0 and x^o is an acute angle; find :

(i) sin x^o (ii) x^o (iii) cos x^o and tan x^o.

Solution 3

(i)

(ii)

(iii)

Question 4

If 4 cos² x^o - 1 = 0 and 0 x^o 90^o, find:

(i) x^o (ii) sin² x^o + cos² x^o

(iii)

Solution 4

(i)

(ii)

(iii)

Question 5

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

Solution 5

Question 6

If sin 3A = 1 and 0 A 90^o, 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 (75^o - A) + cos² (45^o +A)

Solution 7

(i)

(ii)

(iii)

Question 8

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

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

Solution 8

(i)

Given that x = 30^o

(ii)

Given that B = 90^o

Question 9

From the given figure, find:

(i) cos x^o(ii) x^o

(iii)

(iv) Use tan x^o, to find the value of y.

Solution 9

(i)

(ii)

(iii)

(iv)

$W e space k n o w space t h a t space tan x degree equals fraction numerator A B over denominator B C end fraction rightwards double arrow tan x degree equals y over 10 rightwards double arrow y equals 10 tan x degree rightwards double arrow y equals 10 tan 60 degree rightwards double arrow y equals 10 square root of 3$