## NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry Exercise 8

**NCERT Solutions For Class 10 Maths Chapter 8 Introduction to Trigonometry**** Exercise 8 **are prepared by specialised experienced mathematic teacher. Maths are most important subject of board and with the help of this chapter-wise NCERT solution and little practices you can get very good marks in your respective board exam. It also help to build a foundation for topics that will be covered in the upcoming 11th and 12th. Student can also check the **Important Question with solution for class 9 to class 12.**

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**Class 10 Maths Chapter 8 Introduction to Trigonometry Exercise 8** contain total 4 exercise that has 27 questions and it covered the topic trigonometric ratios of the angles, trigonometric ratios for angles of 0^{0} and 90^{0 , }trigonometric ratios of complementary angles. **Check Previous chapter – NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry.**

### Important Formula

**Trigonometric Ratios of Complementary Angles**

sin (90° – A) = cos A,

cos (90° – A) = sin A,

tan (90° – A) = cot A,

cot (90° – A) = tan A,

sec (90° – A) = cosec A,

cosec (90° – A) = sec A

sin^{2} A + cos^{2} A = 1,

sec^{2} A – tan^{2} A = 1 for 0° ≤ A < 90°,

cosec^{2} A = 1 + cot^{2} A for 0° < A ≤ 90°

**Trigonometry Table**

Angle | 0° | 30° | 45° | 60° | 90° |

Sinθ | 0 | 1/2 | 1/√2 | √3/2 | 1 |

Cosθ | 1 | √3/2 | 1/√2 | ½ | 0 |

Tanθ | 0 | 1/√3 | 1 | √3 | Not defined |

Cotθ | Not defined | √3 | 1 | 1/√3 | 0 |

Secθ | 1 | 2/√3 | √2 | 2 | Not defined |

Cosecθ | Not defined | 2 | √2 | 2/√3 | 1 |

#### Trigonometric Ratios

Trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.

Let ∆ABC be a triangle right angled at B. Then the trigonometric ratios of the angle A in right ∆ABC are defined as follows:

### Exercise 8.1

**Question 1.**

**In ∆ABC right angled at B, AB = 24 cm, BC = 7 cm. Determine:**

**(i) sin A, cos A**

**(ii) sin C, cos C**

**Solution:**

**Question 2.**

**In given figure, find tan P – cot R.**

**Solution:**

**Question 3.**

**If sin A = 3/4, Calculate cos A and tan A**

**Solution:**

**Question 4.**

**Given 15 cot A = 8, find sin A and sec A.**

**Solution:**

**Question 5.**

**Given sec θ = 13/12 Calculate all other trigonometric ratios**

**Solution:**

**Question 6.**

**If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.**

**Solution:**

**Question 7.**

**If cot θ = 78, evaluate:**

**(i) (1+𝑠𝑖𝑛𝜃)(1−𝑠𝑖𝑛𝜃)(1+𝑐𝑜𝑠𝜃)(1−𝑐𝑜𝑠𝜃)**

**(ii) cot²θ**

Solution:

**Question 8.**

**If 3 cot A = 4, check whether 1−𝑡𝑎𝑛2𝐴1+𝑡𝑎𝑛2𝐴 = cos² A – sin² A or not.**

**Solution:**

**Question 9.**

**In triangle ABC, right angled at B, if tan A = 1√3, find the value of:**

**(i) sin A cos C + cos A sin C**

**(ii) cos A cos C – sin A sin C**

**Solution:**

**Question 10.**

**In ΔPQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the values of sin P, cos P and tan P.**

**Solution:**

**Question 11.**

**State whether the following statements are true or false. Justify your answer.**

**(i) The value of tan A is always less than 1.**

**(ii) sec A = 125 for some value of angle A.**

**(iii) cos A is the abbreviation used for the cosecant of angle A.**

**(iv) cot A is the product of cot and A.**

**(v) sin θ = 43 for some angle.**

**Solution:**

### Exercise 8.2

**1. Evaluate the following: **

**(i) sin 60° cos 30° + sin 30° cos 60°**

**(ii) 2 tan ^{2} 45° + cos^{2} 30° – sin^{2} 60**

Solution:

**Question 2.**

**Choose the correct option and justify your choice**:

Solution:

**Question 3.**

**If tan (A + B) = √3 and tan (A – B) = 1√3; 0° < A + B ≤ 90°; A > B, find A and B.**

**Solution:**

**Question 4.**

**State whether the following statements are true or false. Justify your answer.**

**(i) sin (A + B) = sin A + sin B.**

**(ii) The value of sin θ increases as θ increases.**

**(iii) The value of cos θ increases as θ increases.**

**(iv) sin θ = cos θ for all values of θ.**

**(v) cot A is not defined for A = 0°.**

**Solution:**

### Exercise 8.3

**Question 1.**

**Evaluate:**

Solution:

**Question 2.**

**(i) tan 48° tan 23° tan 42° tan 67° = 1**

**(ii) cos 38° cos 52° – sin 38° sin 52° = 0**

**Solution:**

**Question 3.**

**If tan 2A = cot (A – 18°), where 2A is an acute angle, find the value of A.**

**Solution:**

**Question 4.**

**If tan A = cot B, prove that A + B = 90°.**

**Solution:**

**Question 5.**

**If sec 4A = cosec (A – 20°), where 4A is an acute angle, find the value of A.**

**Solution:**

**Question 6.**

**If A, B and C are interior angles of a triangle ABC, then show that: sin (𝐵+𝐶2) = cos 𝐴2**

**Solution:**

**Question 7.**

**Express sin 61° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.**

**Solution:**

### Exercise 8.4

**Question 1.**

**Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.**

**Solution:**

**Question 2.**

**Write all the other trigonometric ratios of ∠A in terms of sec A.**

**Solution:**

**Question 3.**

**Evaluate:**

**Solution:**

**Question 4.**

**Choose the correct option. Justify your choice.
(i) 9 sec ^{2}A – 9 tan^{2}A =
(A) 1 (B) 9 (C) 8 (D) 0
(ii) (1 + tan θ + sec θ) (1 + cot θ – cosec θ)
(A) 0 (B) 1 (C) 2 (D) – 1
(iii) (sec A + tan A) (1 – sin A) =
(A) sec A (B) sin A (C) cosec A (D) cos A**

**(iv) 1+tan ^{2}A/1+cot^{2}A = **

** (A) sec ^{2 }A (B) -1 (C) cot^{2}A (D) tan^{2}A**

Solution:

**Question 5.**

**Prove the following identities, where the angles involved are acute angles for which the expressions are defined.**

Solution: