Introduction to Trigonometry Class 10 Notes Maths Chapter 8
Introduction to Trigonometry Class 10 Notes Maths Chapter 8
CBSE Class 10 Maths Notes Chapter 8 Introduction to Trigonometry Pdf free download is part of Class 10 Maths Notes for Quick Revision. Here we have given NCERT Class 10 Maths Notes Chapter 8 Introduction to Trigonometry. According to new CBSE Exam Pattern, MCQ Questions for Class 10 Maths Carries 20 Marks.
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