#### Chapters

Chapter 2: Compound Interest (Without using formula)

Chapter 3: Compound Interest (Using Formula)

Chapter 4: Expansions (Including Substitution)

Chapter 5: Factorisation

Chapter 6: Simultaneous (Linear) Equations (Including Problems)

Chapter 7: Indices (Exponents)

Chapter 8: Logarithms

Chapter 9: Triangles [Congruency in Triangles]

Chapter 10: Isosceles Triangles

Chapter 11: Inequalities

Chapter 12: Mid-point and Its Converse [ Including Intercept Theorem]

Chapter 13: Pythagoras Theorem [Proof and Simple Applications with Converse]

Chapter 14: Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]

Chapter 15: Construction of Polygons (Using ruler and compass only)

Chapter 16: Area Theorems [Proof and Use]

Chapter 17: Circle

Chapter 18: Statistics

Chapter 19: Mean and Median (For Ungrouped Data Only)

Chapter 20: Area and Perimeter of Plane Figures

Chapter 21: Solids [Surface Area and Volume of 3-D Solids]

Chapter 22: Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and their Reciprocals]

Chapter 23: Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios]

Chapter 24: Solution of Right Triangles [Simple 2-D Problems Involving One Right-angled Triangle]

Chapter 25: Complementary Angles

Chapter 26: Co-ordinate Geometry

Chapter 27: Graphical Solution (Solution of Simultaneous Linear Equations, Graphically)

Chapter 28: Distance Formula

## Chapter 23: Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios]

### Selina solutions for Concise Mathematics Class 9 ICSE Chapter 23 Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios] Exercise 23 (A) [Pages 291 - 292]

find the value of: sin 30° cos 30°

find the value of: tan 30° tan 60°

find the value of: cos^{2} 60° + sin^{2} 30°

find the value of: cosec^{2} 60° - tan^{2} 30°

find the value of: sin^{2} 30° + cos^{2} 30°+ cot^{2} 45°

find the value of: cos^{2} 60° + sec^{2} 30° + tan^{2} 45°

**find the value of :**tan

^{2}30° + tan

^{2}45° + tan

^{2}60°

**find the value of :**

`( tan 45°)/ (cos ec30°) +( sec60°)/(co 45°) – (5 sin 90°)/ (2 cos 0°)`

**find the value of :**

3sin^{2} 30° + 2tan^{2} 60° - 5cos^{2} 45°

**Prove that:**

sin 60° cos 30° + cos 60° . sin 30° = 1

**Prove that:**

cos 30° . cos 60° - sin 30° . sin 60° = 0

**Prove that:**

cosec^{2} 45° - cot^{2} 45° = 1

**Prove that:**

cos^{2} 30° - sin^{2} 30° = cos 60°

**Prove that:**

`((tan60° + 1)/(tan 60° – 1))^2 = (1+ cos 30°) /(1– cos 30°) `

**Prove that:**

3 cosec^{2} 60° - 2 cot^{2} 30° + sec^{2} 45° = 0

**prove that:**

sin (2 x 30°) = `(2 tan 30°)/(1+tan^2 30°)`

**prove that:**

cos (2 x 30°) = `(1 – tan^2 30°)/(1+tan^2 30°)`

**prove that:**

tan (2 x 30°) = `(2 tan 30°)/(1– tan^2 30°)`

ABC is an isosceles right-angled triangle. Assuming of AB = BC = x, find the value of each of the following trigonometric ratios: sin 45°

ABC is an isosceles right-angled triangle. Assuming of AB = BC = x, find the value of each of the following trigonometric ratio: cos 45°

ABC is an isosceles right-angled triangle. Assuming of AB = BC = x, find the value of each of the following trigonometric ratios: tan 45°

**Prove that:**sin 60° = 2 sin 30° cos 30°

**Prove that:**4 (sin

^{4}30° + cos

^{4}60°) -3 (cos

^{2}45° - sin

^{2}90°) = 2

If sin x = cos x and x is acute, state the value of x

If sec A = cosec A and 0° ∠A ∠90°, state the value of A

If tan θ = cot θ and 0°∠θ ∠90°, state the value of θ

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

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

True

False

secθ . Cot θ= cosecθ ; write true or false

True

False

For any angle θ, state the value of : sin^{2 }θ + cos^{2 }θ

State for any acute angle θ whether sin θ increases or decreases as θ increases

Increases

Decreases

State for any acute angle θ whether cos θ increases or decreases as θ increases.

Increases

Decreases

State for any acute angle θ whether tan θ increases or decreases as θ decreases.

Increases

Decreases

If `sqrt3` = 1.732, find (correct to two decimal place) the value of sin 60^{o}

If `sqrt3` = 1.732, find (correct to two decimal place) the value of `(2)/(tan 30°)`

**Evaluate : **

`(cos3"A" – 2cos4"A")/(sin3"A" + 2sin4"A")` , when A = 15°

**Evaluate : **

`(3 sin 3"B" + 2 cos(2"B" + 5°))/(2 cos 3"B" – sin (2"B" – 10°)` ; when "B" = 20°.

### Selina solutions for Concise Mathematics Class 9 ICSE Chapter 23 Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios] Exercise 23 (B) [Page 293]

Given A = 60° and B = 30°, **prove that : **sin (A + B) = sin A cos B + cos A sin B

Given A = 60° and B = 30°, **prove that : **cos (A + B) = cos A cos B - sin A sin B

Given A = 60° and B = 30°, **prove that : **cos (A - B) = cos A cos B + sin A sin B

Given A = 60° and B = 30°, **prove that :** tan (A - B) = `(tan"A" – tan"B")/(1 + tan"A".tan"B")`

If A =30^{o}, then prove that :

sin 2A = 2sin A cos A = `(2 tan"A")/(1 + tan^2"A")`

If A =30^{o}, then prove that :

cos 2A = cos^{2}A - sin^{2}A = `(1 – tan^2"A")/(1+ tan^2"A")`

If A =30^{o}, then prove that :

2 cos^{2} A - 1 = 1 - 2 sin^{2}A

If A =30^{o}, then prove that :

sin 3A = 3 sin A - 4 sin^{3}A.

If A = B = 45° , **show that:**sin (A - B) = sin A cos B - cos A sin B

If A = B = 45° , **show that:**cos (A + B) = cos A cos B - sin A sin B

If A = 30°; **show that:**

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

If A = 30°; **show that:**

(sinA - cosA)^{2} = 1 - sin2A

If A = 30°; **show that:**

cos 2A = cos^{4} A - sin^{4} A

If A = 30°; **show that:**

`(1 – cos 2"A")/(sin 2"A") = tan"A"`

If A = 30°; **show that:**

`(1 + sin 2"A" + cos 2"A")/(sin "A" + cos"A") = 2 cos "A"`

If A = 30°; **show that:**

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

If A = 30°; **show that:**

`(cos^3"A" – cos 3"A")/(cos "A") + (sin^3"A" + sin3"A")/(sin"A") = 3`

### Selina solutions for Concise Mathematics Class 9 ICSE Chapter 23 Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios] Exercise 23 (C) [Pages 297 - 298]

Solve the following equation for A, if 2 sin A = 1

Solve the following equation for A, if 2cos2A = 1

Solve the following equation for A, if sin 3 A = `sqrt3 /2`

Solve the following equation for A, if sec 2A = 2

Solve the following equations for A, if tan A = 1

Solve the following equation for A, if `sqrt3` tan 3 A = 1

Solve the following equation for A, if 2 sin 3 A = 1

Solve the following equation for A, if `sqrt3` cot 2 A = 1

Calculate the value of A, if (sin A - 1) (2 cos A - 1) = 0

Calculate the value of A, if (tan A - 1) (cosec 3A - 1) = 0

Calculate the value of A, if (sec 2A - 1) (cosec 3A - 1) = 0

Calculate the value of A, if cos 3A. (2 sin 2A - 1) = 0

Calculate the value of A, if (cosec 2A - 2) (cot 3A - 1) = 0

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

(i) sin x°

(ii) x°

(iii) cos x and tan x°.

If 4 cos^{2} x° - 1 = 0 and 0 ∠ x° ∠ 90°,

find:(i) x°

(ii) sin^{2} x° + cos^{2} x°

(iii) `(1)/(cos^2xx°) – (tan^2 xx°)`

If 4 sin^{2} θ - 1= 0 and angle θ is less than 90°, find the value of θ and hence the value of cos^{2} θ + tan^{2}θ.

If sin 3A = 1 and 0 < A < 90^{°}, find sin A

If sin 3A = 1 and 0 < A < 90^{°}, find cos 2A

If sin 3A = 1 and 0 < A < 90^{°}, find `tan^2A - (1)/(cos^2 "A")`

If 2 cos 2A = `sqrt3` and A is acute,

find:

(i) A

(ii) sin 3A

(iii) sin^{2} (75° - A) + cos^{2} (45° +A)

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

If 3 tan A - 5 cos B = `sqrt3` and B = 90°, find the value of A

From the given figure, **find:**

(i) cos x°^{}(ii) x°^{}(iii) `(1)/(tan^2 xx°) – (1)/(sin^2xx°)`

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

**Use the given figure to find:**(i) tan

^{ }θ°

(ii)

^{ }θ°

(iii) sin

^{2}θ° - cos

^{2}θ°

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

Find the magnitude of angle A, if 2 sin A cos A - cos A - 2 sin A + 1 = 0

Find the magnitude of angle A, if tan A - 2 cos A tan A + 2 cos A - 1 = 0

Find the magnitude of angle A, if 2 cos^{2} A - 3 cos A + 1 = 0

Find the magnitude of angle A, if 2 tan 3A cos 3A - tan 3A + 1 = 2 cos 3A

Solve for x : 2 cos 3x - 1 = 0

Solve for x : cos `(x)/(3) –1` = 0

Solve for x : sin (x + 10°) = `(1)/(2)`

Solve for x : cos (2x - 30°) = 0

Solve for x : 2 cos (3x - 15°) = 1

Solve for x : tan^{2} (x - 5°) = 3

Solve for x : 3 tan^{2} (2x - 20°) = 1

Solve for x : cos `(x/(2)+10°) = (sqrt3)/(2)`

Solve for x : sin^{2} x + sin^{2} 30° = 1

Solve for x : cos^{2} 30° + cos^{2} x = 1

Solve for x : cos^{2} 30° + sin^{2} 2x = 1

Solve for x : sin^{2} 60° + cos^{2} (3x- 9°) = 1

If 4 cos^{2} x = 3 and x is an acute angle; **find the value of :**

(i) x

(ii) cos^{2} x + cot^{2} x

(iii) cos 3x (iv) sin 2x

In ΔABC, ∠B = 90° , AB = y units, BC = `(sqrt3)` units, AC = 2 units and angle A = x°,

find:

(i) sin x°

(ii) x°

(iii) tan x°

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

If 2 cos (A + B) = 2 sin (A - B) = 1;

find the values of A and B.

## Chapter 23: Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios]

## Selina solutions for Concise Mathematics Class 9 ICSE chapter 23 - Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios]

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Concepts covered in Concise Mathematics Class 9 ICSE chapter 23 Trigonometrical Ratios of Standard Angles [Including Evaluation of an Expression Involving Trigonometric Ratios] are Trigonometric Equation Problem and Solution, Trigonometric Ratios of Some Special Angles, Trigonometric Ratios of Some Special Angles, Trigonometric Ratios of Some Special Angles.

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