## CBSE Class 10 Maths Notes Chapter 8 Introduction to Trigonometry

Introduction to Trigonometry:

Trigonometry is the branch of mathematics that deals with the study of relationships between the sides and angles of triangles. It is a very important topic for students studying in Class 10 Maths. It is used in various fields such as physics, engineering, and even in day-to-day life.

In Class 10 Maths, the basic concepts of trigonometry are introduced. Some of the key topics covered in this chapter are:

- Trigonometric ratios of an acute angle
- Trigonometric ratios of complementary angles
- Trigonometric identities
- Heights and distances

The trigonometric ratios of an acute angle are defined as the ratios of the sides of a right-angled triangle. There are three basic ratios:

- Sine (sin) ratio = opposite side / hypotenuse
- Cosine (cos) ratio = adjacent side / hypotenuse
- Tangent (tan) ratio = opposite side / adjacent side

The other three ratios are the reciprocals of the above ratios:

- Cosecant (cosec) ratio = hypotenuse / opposite side
- Secant (sec) ratio = hypotenuse / adjacent side
- Cotangent (cot) ratio = adjacent side / opposite side

The trigonometric ratios of complementary angles are useful in solving problems related to angles that add up to 90 degrees. The sum of the sine of an angle and its complementary angle is always equal to 1.

Trigonometric identities are equations that are true for all values of the variables. Some of the important identities are:

- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ

Heights and distances are also a part of trigonometry. In this topic, we learn how to find the height of an object or the distance between two objects, using the concepts of trigonometry.

Overall, trigonometry is an important chapter in Class 10 Maths, as it forms the foundation for higher-level mathematics and other subjects such as physics and engineering.

**CBSE Class 10 Maths Important Questions Chapter 8 Introduction to Trigonometry**

- Define trigonometric ratios.

Trigonometric ratios are defined as the ratios of the sides of a right-angled triangle with respect to its acute angles. There are three primary trigonometric ratios: sine, cosine, and tangent.

- State the Pythagorean theorem.

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

- What is the range of values of sine and cosine ratios?

The values of sine and cosine ratios range from -1 to 1, inclusive.

- What is the relation between cosecant and sine?

The cosecant of an angle is the reciprocal of its sine. That is, if sin A = x, then csc A = 1/x.

- What is the relation between tangent and cotangent?

The tangent of an angle is the reciprocal of its cotangent. That is, if cot A = x, then tan A = 1/x.

- How can we find the value of a trigonometric ratio if we know the value of another ratio?

We can use the following trigonometric identities to find the value of a trigonometric ratio if we know the value of another ratio:

- sin^2A + cos^2A = 1
- 1 + tan^2A = sec^2A
- 1 + cot^2A = csc^2A

- What is the use of trigonometry in real life?

Trigonometry is used in a wide range of real-life applications, including:

- Calculating the height and distance of buildings, mountains, and other objects
- Navigation and surveying
- Astronomy and space exploration
- Engineering and construction
- Music and sound waves
- Medicine and biology

- What is the difference between acute and obtuse angles?

An acute angle is an angle that measures less than 90 degrees, while an obtuse angle measures more than 90 degrees but less than 180 degrees.

- What is the complementary angle of an angle?

The complementary angle of an angle is the angle that, when added to the original angle, results in a sum of 90 degrees.

- What is the supplementary angle of an angle?

The supplementary angle of an angle is the angle that, when added to the original angle, results in a sum of 180 degrees.

**CBSE Class 10 Maths Important Questions Answers Chapter 8 Introduction to Trigonometry**

- What is the value of sin 60° + cos 30°?

Answer: sin 60° = √3/2 and cos 30° = √3/2 Therefore, sin 60° + cos 30° = √3/2 + √3/2 = √3

- Find the value of sin 30°.

Answer: sin 30° = 1/2

- Find the value of cos 60°.

Answer: cos 60° = 1/2

- If sin A = 3/5, find cos A.

Answer: We know that sin²A + cos²A = 1 Therefore, cos²A = 1 – sin²A = 1 – (3/5)² = 1 – 9/25 = 16/25 Hence, cos A = ±√(16/25) = ±4/5 Since A is acute, cos A = 4/5

- Find the value of tan 45°.

Answer: tan 45° = 1

- Find the value of sin²θ + cos²θ.

Answer: sin²θ + cos²θ = 1 (as per the trigonometric identity sin²θ + cos²θ = 1)

- Find the value of tan A if sin A = 4/5 and cos A = 3/5.

Answer: We know that tan A = sin A/cos A = (4/5)/(3/5) = 4/3

- If sin A = x and cos A = y, find the value of x² + y².

Answer: We know that sin²A + cos²A = 1 Therefore, x² + y² = 1

- If tan A = 1/√3, find the value of sin A and cos A.

Answer: We know that tan A = sin A/cos A Therefore, sin A/cos A = 1/√3 Squaring both sides, we get (sin A)²/(cos A)² = 1/3 (sin A)² = (cos A)²/3 (sin A)² + (cos A)² = 1 4(cos A)²/3 = 1 (cos A)² = 3/4 cos A = ±√(3/4) = ±√3/2 Since A is acute, cos A = √3/2 sin A = (sin A/cos A) × cos A = (1/√3) × (√3/2) = 1/2

- If sec θ = 2, find the value of cos θ.

Answer: We know that sec θ = 1/cos θ Therefore, 1/cos θ = 2 cos θ = 1/2

- If cos 2θ = 1/2, find the value of sin 2θ.

Answer: We know that cos 2θ = 2(cos θ)² – 1 Therefore, 2(cos θ)² – 1 = 1/2 (cos θ)² = 3/4 cos θ = ±√(3/4) = ±√3/2 Since 2θ is acute, cos 2θ = 2(cos θ)² – 1 = 2(3/4) – 1 = 1/2 sin

**CBSE Class 10 Maths Important Questions Answers MCQs Chapter 8 Introduction to Trigonometry**

**CBSE Class 10 Maths Important Questions Answers MCQs Chapter 8 Introduction to Trigonometry**

- In a right triangle ABC, if angle A is 30°, then what is the value of sin B? A) 1/2 B) 1/√2 C) √3/2 D) √2/2 Answer: A) 1/2
- In a right triangle ABC, if sin A = 3/5, then what is the value of cos A? A) 4/5 B) 3/4 C) √2/5 D) √2/3 Answer: A) 4/5
- In a right triangle ABC, if tan A = 1/3, then what is the value of sin A? A) 1/√10 B) 3/√10 C) √10/3 D) √10/4 Answer: B) 3/√10
- If sin θ = 3/5 and cos θ < 0, then what is the value of θ in quadrant III? A) 208.68° B) 151.05° C) 331.31° D) 28.69° Answer: C) 331.31°
- If sin θ = 4/5 and cos θ < 0, then what is the value of θ in quadrant II? A) 53.13° B) 126.87° C) 233.13° D) 306.87° Answer: C) 233.13°
- If sin A = 1/2 and cos B = 1/√3, then what is the value of tan (A + B)? A) √3/3 B) 1/√3 C) √3/2 D) 2/√3 Answer: A) √3/3
- If tan A = 4/3 and tan B = 5/12, then what is the value of tan (A – B)? A) 47/39 B) 39/47 C) 12/5 D) 3/4 Answer: B) 39/47
- If sin A = x and sin B = y, then what is the value of cos (A + B) in terms of x and y? A) xy + √(1 – x²)(1 – y²) B) xy – √(1 – x²)(1 – y²) C) √(1 – x²)(1 – y²) – xy D) xy/√(1 – x²)(1 – y²) Answer: A) xy + √(1 – x²)(1 – y²)
- If sin A = 3/5 and cos B = 5/13, then what is the value of tan (A – B)? A) 52/23 B) 23/52 C) 8/15 D) 15/8 Answer: A) 52/23
- If sin θ = x and cos θ = y, then what is the value of tan θ in terms of x and y? A) xy B) x/y C) y/x D) (y-x)/(1+xy) Answer: B) x/y

## CBSE Class 10 Maths Notes Chapter 8 Introduction to Trigonometry

- Position of a point P in the Cartesian plane with respect to co-ordinate axes is represented by the ordered pair (x, y).
- Trigonometry is the science of relationships between the sides and angles of a right-angled triangle.
- Trigonometric Ratios: Ratios of sides of right triangle are called trigonometric ratios.

Consider triangle ABC right-angled at B. These ratios are always defined with respect to acute angle ‘A’ or angle ‘C. - If one of the trigonometric ratios of an acute angle is known, the remaining trigonometric ratios of an angle can be easily determined.
- How to identify sides: Identify the angle with respect to which the t-ratios have to be calculated. Sides are always labelled with respect to the ‘θ’ being considered.

Let us look at both cases:

In a right triangle ABC, right-angled at B. Once we have identified the sides, we can define six t-Ratios with respect to the sides.

case I | case II |

(i) sine A = perpendicularhypotenuse=BCAC | (i) sine C = perpendicularhypotenuse=ABAC |

(ii) cosine A = basehypotenuse=ABAC | (ii) cosine C = basehypotenuse=BCAC |

(iii) tangent A = perpendicularbase=BCAB | (iii) tangent C = perpendicularbase=ABBC |

(iv) cosecant A = hypotenuseperpendicular=ACBC | (iv) cosecant C = hypotenuseperpendicular=ACAB |

(v) secant A = hypotenusebase=ACAB | (v) secant C = hypotenusebase=ACBC |

(v) cotangent A = baseperpendicular=ABBC | (v) cotangent C = baseperpendicular=BCAB |

Note from above six relationships:

cosecant A = 1sinA, secant A = 1cosineA, cotangent A = 1tanA,

However, it is very tedious to write full forms of t-ratios, therefore the abbreviated notations are:

sine A is sin A

cosine A is cos A

tangent A is tan A

cosecant A is cosec A

secant A is sec A

cotangent A is cot A

**TRIGONOMETRIC IDENTITIES**

An equation involving trigonometric ratio of angle(s) is called a trigonometric identity, if it is true for all values of the angles involved. These are:

tan θ = sinθcosθ

cot θ = cosθsinθ

- sin² θ + cos² θ = 1 ⇒ sin² θ = 1 – cos² θ ⇒ cos² θ = 1 – sin² θ
- cosec² θ – cot² θ = 1 ⇒ cosec² θ = 1 + cot² θ ⇒ cot² θ = cosec² θ – 1
- sec² θ – tan² θ = 1 ⇒ sec² θ = 1 + tan² θ ⇒ tan² θ = sec² θ – 1
- sin θ cosec θ = 1 ⇒ cos θ sec θ = 1 ⇒ tan θ cot θ = 1

**ALERT:**

A t-ratio only depends upon the angle ‘θ’ and stays the same for same angle of different sized right triangles.**Value of t-ratios of specified angles:**

∠A | 0° | 30° | 45° | 60° | 90° |

sin A | 0 | 12 | 12√ | 3√2 | 1 |

cos A | 1 | 3√2 | 12√ | 12 | 0 |

tan A | 0 | 13√ | 1 | √3 | not defined |

cosec A | not defined | 2 | √2 | 23√ | 1 |

sec A | 1 | 23√ | √2 | 2 | not defined |

cot A | not defined | √3 | 1 | 13√ | 0 |

The value of sin θ and cos θ can never exceed 1 (one) as opposite side is 1. Adjacent side can never be greater than hypotenuse since hypotenuse is the longest side in a right-angled ∆.

**‘t-RATIOS’ OF COMPLEMENTARY ANGLES**

If ∆ABC is a right-angled triangle, right-angled at B, then

∠A + ∠C = 90° [∵ ∠A + ∠B + ∠C = 180° angle-sum-property]

or ∠C = (90° – ∠A)

**Thus, ∠A and ∠C are known as complementary angles and are related by the following relationships:**

sin (90° -A) = cos A; cosec (90° – A) = sec A

cos (90° – A) = sin A; sec (90° – A) = cosec A

tan (90° – A) = cot A; cot (90° – A) = tan A

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