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 = \(\frac { perpendicular }{ hypotenuse } =\frac { BC }{ AC } \)	(i) sine C = \(\frac { perpendicular }{ hypotenuse } =\frac { AB }{ AC } \)
(ii) cosine A = \(\frac { base }{ hypotenuse } =\frac { AB }{ AC } \)	(ii) cosine C = \(\frac { base }{ hypotenuse } =\frac { BC }{ AC } \)
(iii) tangent A = \(\frac { perpendicular }{ base } =\frac { BC }{ AB } \)	(iii) tangent C = \(\frac { perpendicular }{ base } =\frac { AB }{ BC } \)
(iv) cosecant A = \(\frac { hypotenuse }{ perpendicular } =\frac { AC }{ BC } \)	(iv) cosecant C = \(\frac { hypotenuse }{ perpendicular } =\frac { AC }{ AB } \)
(v) secant A = \(\frac { hypotenuse }{ base } =\frac { AC }{ AB } \)	(v) secant C = \(\frac { hypotenuse }{ base } =\frac { AC }{ BC } \)
(v) cotangent A = \(\frac { base }{ perpendicular } =\frac { AB }{ BC } \)	(v) cotangent C = \(\frac { base }{ perpendicular } =\frac { BC }{ AB } \)

Note from above six relationships:

cosecant A = \(\frac { 1 }{ sinA }\), secant A = \(\frac { 1 }{ cosineA }\), cotangent A = \(\frac { 1 }{ tanA }\),

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 = \(\frac { sin\theta }{ cos\theta } \)
cot θ = \(\frac { cos\theta }{ sin\theta } \)

• 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	\(\frac { 1 }{ 2 }\)	\(\frac { 1 }{ \sqrt { 2 } } \)	\(\frac { \sqrt { 3 } }{ 2 } \)	1
cos A	1	\(\frac { \sqrt { 3 } }{ 2 } \)	\(\frac { 1 }{ \sqrt { 2 } } \)	\(\frac { 1 }{ 2 }\)	0
tan A	0	\(\frac { 1 }{ \sqrt { 3 } } \)	1	√3	not defined
cosec A	not defined	2	√2	\(\frac { 2 }{ \sqrt { 3 } } \)	1
sec A	1	\(\frac { 2 }{ \sqrt { 3 } } \)	√2	2	not defined
cot A	not defined	√3	1	\(\frac { 1 }{ \sqrt { 3 } } \)	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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