Selina solutions concise maths class 9 icse Solutions for Chapter 23 chapter 23 trigonometrical ratios of standard angles

Exercises in Chapter 23 chapter 23 trigonometrical ratios of standard angles Grade 9

Exercise 23(A)

Exercise 23(B)

Exercise 23(C)

Questions in Exercise 23(A)

Find the value of:

(i) sin 30 ° cos 30°

(ii) tan 30° tan 60°

(iii) cos2 60° + sin2 30°

(iv) cosec2 60° -tan2 30°

(v) sin2 30° + cos2 30° + cot2 45°

(vi) cos2 60° + sec2 30° + tan2 45° .

Find the value of :

(i) tan2 30 ° + tan2 45° + tan2 60°

(ii) Question 2 Image - Selina Solutions CONCISE Maths - Class 9 ICSE chapter Chapter 23- Trigonometrical Ratios of Standard Angles

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

Prove that:

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

(ii) cos 30° . cos 60° -sin 30° . sin 60° = 0

(iii) cosec2 45° -cot2 45° = 1

(iv) cos2 30° -sin2 30° = cos 60° .

(v) Question 3 Image - Selina Solutions CONCISE Maths - Class 9 ICSE chapter Chapter 23- Trigonometrical Ratios of Standard Angles

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

Prove that:

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 °

(ii) cos 45°

(iii) tan 45°

Question 5 Image - Selina Solutions CONCISE Maths - Class 9 ICSE chapter Chapter 23- Trigonometrical Ratios of Standard Angles

Prove that:

(i) sin 60 ° = 2 sin 30° cos 30° .

(ii) 4 (sin4 30° + cos4 60° )-3 (cos2 45° -sin2 90° ) = 2

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

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

(iii) If tanθ = cotθ and 0° <= θ <= 90° , state the value of θ.

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

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

(ii) secθ . Cotθ = cosecθ ; write true or false.

(iii) For any angle, state the value of : Sin2θ + cos2θ .

State for any acute angle 𝜃 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.

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

(i) sin 60 °

(ii) 2/tan 30°

Evaluate:

Questions in Exercise 23(B)

Given A = 60 ° and B = 30° , 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 1 Image - Selina Solutions CONCISE Maths - Class 9 ICSE chapter Chapter 23- Trigonometrical Ratios of Standard Angles

If A =30 ° , then prove that:

(i) sin 2A = 2sin A cos A = 𝟐𝒕𝒂𝒏𝑨/ 𝟏+𝒕𝒂𝒏𝟐𝑨

(ii) cos 2A = cos2A - sin2A= 𝟏−𝒕𝒂𝒏𝟐𝑨/ 𝟏+𝒕𝒂𝒏𝟐𝑨

(iii) 2 cos2 A - 1 = 1 - 2 sin2A

(iv) sin 3A = 3 sin A - 4 sin3A

If A = B = 45 ° , 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

If A = 30 ° ; show that:

(i) sin 3 A = 4 sin A sin (60° -A) sin (60° + A)

(ii) (sin A - cos A)2 = 1 - sin 2A

(iii) cos 2A = cos4 A - sin4 A

(iv) 𝟏−𝒄𝒐𝒔 𝟐𝑨/ 𝒔𝒊𝒏 𝟐𝑨 = 𝒕𝒂𝒏𝑨

(v) 𝟏+𝒔𝒊𝒏𝟐𝑨+𝒄𝒐𝒔𝟐𝑨/ 𝒔𝒊𝒏𝑨+𝒄𝒐𝒔 𝑨 = 𝟐𝒄𝒐𝒔 𝑨

(vi) 4 cos A cos (60° -A). cos (60° + A) = cos 3A

(vii) 𝑐𝑜𝑠3𝐴−𝑐𝑜𝑠3𝐴/ 𝑐𝑜𝑠𝐴

𝑠𝑖𝑛3𝐴−𝑠𝑖𝑛3𝐴/ 𝑠𝑖𝑛𝐴 = 3

Questions in Exercise 23(C)

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

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

If 2 sin x ° -1 = 0 and xo is an acute angle; find :

(i) sin x °

(ii) x °

(iii) cos x ° and tan xo .

If 4 cos2 x ° -1 = 0 and 0<= x ° <=90° , find:

(i) x °

(ii) sin2 x ° + cos2 x °

(iii) 𝟏/ 𝒄𝒐𝒔𝟐𝒙° − 𝒕𝒂𝒏𝟐𝒙°

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

If sin 3A = 1 and 0 <= A <= 90 ° , find:

(i) sin A

(ii) cos 2A

(iii) 𝑡𝑎𝑛2𝐴 − 1/ 𝑐𝑜𝑠2A

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

(i) A

(ii) sin 3A

(iii) sin2 (75 ° -A) + cos2 (45° +A)

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

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

From the given figure, find:

(i) cos x °

(ii) x °

(iii) 𝟏 𝒕𝒂𝒏𝟐𝒙° − 𝟏 𝒔𝒊𝒏𝟐𝒙°

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

Question 9 Image - Selina Solutions CONCISE Maths - Class 9 ICSE chapter Chapter 23- Trigonometrical Ratios of Standard Angles

Use the given figure to find:

(i) tan 𝜽°

(ii) 𝜽°

(iii) sin2 𝜽 °

-cos2 𝜽°

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

Question 10 Image - Selina Solutions CONCISE Maths - Class 9 ICSE chapter Chapter 23- Trigonometrical Ratios of Standard Angles

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

Solve for x:

(i) 2 cos 3x - 1 = 0

(ii) Cos 𝒙/ 𝟑 − 𝟏 = 0

(iii) sin (x + 10o ) = ½

(iv) cos (2x - 30 ° ) = 0

(v) 2 cos (3x - 15° ) = 1

(vi) tan2 (x - 5 ° ) = 3

(vii) 3 tan2 (2x - 20° ) = 1

(viii) Cos ( 𝒙/ 𝟐 +𝟏𝟎) = √𝟑/ 𝟐

(ix) sin2 x + sin2 30° = 1

(x) cos2 30° + cos2 x = 1

(xi) cos2 30° + sin2 2x = 1

(xii) sin2 60° + cos2 (3x- 9 ° ) = 1

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

In triangle ABC, angle B = 90 ° , AB = y units, BC = √𝟑 units, AC = 2 units and angle A = xo , find:

(i) sin x°

(ii) x °

(iii) tan x°

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

Question 14 Image - Selina Solutions CONCISE Maths - Class 9 ICSE chapter Chapter 23- Trigonometrical Ratios of Standard Angles

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

Was This helpful?

Chapters in this book

Chapter 1- Rational and Irrational Numbers

Chapter 2- Compound Interest [Without Using Formula

Chapter 3- Compound Interest [Using Formula

Chapter 4- Expansions

Chapter 5- Factorisation

Chapter 6- Simultaneous Equations

Chapter 7- Indices [exponents]

Chapter 8- Logarithms

Chapter 9- Triangles [Congruency in Triangles]

Chapter 10- Isosceles Triangle

Chapter 11- Inequalities

Chapter 12- Mid-Point and Its Converse

Chapter 13- Pythagoras Theorem

Chapter 14- Rectilinear Figures

Chapter 15- Construction of Polygons

Chapter 16- Area Theorems

Chapter 17- Circles

Chapter 18- Statistics

Chapter 19- Mean and Median

Chapter 20- Area and Perimeter of Plane Figures

Chapter 21- Solids

Chapter 22- Trigonometrical Ratios

Chapter 23- Trigonometrical Ratios of Standard Angles

Chapter 24- Solution Of Right Triangles

Chapter 25- Complementary angles

Chapter 26- Co-ordinate Geometry

Chapter 27- Graphical Solution

Chapter 28- Distance Formula