Introduction to Trigonometry: Class 10 Mathematics NCERT Chapter 8

Key Features of NCERT Material for Class 10 Mathematics Chapter 8 – Introduction to Trigonometry

In the last chapter 7, you learned about Coordinate Geometry. In this chapter you will be learning about Introduction to Trigonometry.

Position of a point P in the Cartesian plane as for coordinate axes is shown by the arranged pair (x, y). 

Trigonometry is the science of connections between the sides and angles of a right triangle. 

Trigonometric Ratios: Ratios of sides of the right-angled triangle are called trigonometric ratios. 

Consider right triangle ABC angled at B. These ratios are constantly characterized as for acute angle ‘A’ or angle ‘C’. 

In the event that one of the trigonometric ratios of an acute angle is known, the rest of trigonometric ratios of an angle can be handily decided. 

Instructions to distinguish sides: Identify the point concerning which the t-ratios must be determined. Sides are constantly marked concerning the ‘θ’ being taken into consideration. 

Let us take a look at the two cases: 

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In a right-angled triangle ABC,  angled at B. When we have recognized the sides, we can characterize six t-Ratios as for the sides. 

2-17

Note from over six connections: 

cosecant A = , secant A = , cotangent A =

In any case, it is dreary to compose full types of t-ratios, in this way the contracted documentations 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 including trigonometric ratio of angle(s) is known as a trigonometric identity, in the event that it is valid for all values of the angles in question. These are: 

tan θ =  3-15

Cot θ = 4-16

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 

Note: 

A t-proportion just relies on the point ‘θ’ and remains the equivalent for the same angle of various measured right triangles. 

Value of t-ratios of indicated angles: 

5-10

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The value of tan θ and cos θ can never surpass 1 (one) as the inverse side is 1. Adjacent side can never be more noteworthy than hypotenuse since hypotenuse is the longest side in a right-calculated ∆. 

‘t-RATIOS’ OF COMPLEMENTARY ANGLES 

On the off chance that ∆ABC is a right triangle, angled at B, at that point 

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

or then again ∠C = (90° – ∠A) 

Subsequently, ∠A and ∠C are known as complementary angles and are connected by the accompanying connections: 

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

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

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

Representation of Trigonometric Ratios Using a Unit Circle 

Draw a circle of unit range with the origin as the middle. Consider a line segment OP joining a point P on the circle to the inside which makes an edge θ with the x-axis. Draw an opposite from P to the x-axis to cut it at Q. 

sinθ=PQ/OP=PQ/1=PQ 

cosθ=OQ/OP=OQ/1=OQ 

tanθ=PQ/OQ=sinθ/cosθ 

cosecθ=OP/PQ=1/PQ 

secθ=OP/OQ=1/OQ 

cotθ=OQ/PQ=cosθ/sinθ

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