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

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

Solution 1(a)

Correct option: (i) sin 60o

Solution 1(b)

Correct option: (ii) 20o

Solution 1(c)

Correct option: (iii) 45o

Solution 1(d)

Correct option: (iii) 3

Solution 1(e)

Correct option: (i)

Solution 2

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Solution 3

(i)

(ii)

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

 

Solution 4

(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

Solution 5

(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

Solution 6

Given that AB = BC = x

(i)

(ii)

(iii)

Solution 7

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

Solution 8

(i)

The angle, x is acute and hence we have, 0 < x

 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

Solution 9

(i)

if x and y are acute angles,

is false.

 

(ii)

 


Sec. Cot = cosec is true


(iii)


Solution 10

(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.

 

Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios] Exercise Ex. 23(B)

Solution 1(a)

Correct option: (i) cot A

Solution 1(b)

Correct option: (i) cot A

Solution 1(c)

Correct option: (iii) sin 3A

Solution 1(d)

Correct option: (iv) cos 60o

Solution 1(e)

Correct option: (i) 1

Solution 2

Given A = 60o and B = 30o

(i)

(ii)

(iii)

(iv)

Solution 3

Given A=

(i)

(ii)

(iii)

(iv)

Solution 4

Given that A = B = 45o

(i)

(ii)

Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios] Exercise Ex. 23(C)

Solution 1(a)

Correct option: (i) 30o

Solution 1(b)

Correct option: (i) 90o or 0o

Solution 1(c)

Correct option: (ii) 45o

Solution 1(d)

Correct option: (iii) 10o

Solution 1(e)

Correct option: (iv) 30o

Solution 2

(i)

(ii)

(iii)

(iv)

(V)

(vi)

(vii)

(viii)

Solution 3

(i)

(ii)

(iii)

(iv)

(v)

Solution 4

(i)

 

(ii)

(iii)

Solution 5

(i)

(ii)

(iii)

Solution 6

Solution 7

(i)

(ii)

(iii)

Solution 8

(i)

 

(ii)

(iii)

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

Solution 1

(i)

(ii)

Solution 2

(i) Given that A=

(ii) Given that B=

Solution 3

Given that A = 30o

(i)

 

(ii)

 

(iii)

(iv)

(v)

(vi)

(vii)

 

Solution 4

(i)

Given that x = 30o

 

(ii)

Given that B = 90o

Solution 5

(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

Solution 6

(i)

(ii)

(iii)

(iv)

Solution 7

(i)

 

(ii)

(iii)

(iv)

Solution 8

(i)

 

(ii)

 

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii)

 

Solution 9

(i)

(ii)

(iii)

(iv)

Solution 10

(i)

From

(ii)

(iii)

(iv)

Solution 11

Adding (1) and (2)

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