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S.L. Loney General Expressions for Trigonometrical Ratio Solutions (Chapter 6)

SL Loney Plane Trigonometry Solutions Chapter 6 ‘General Expressions for All Angles Having a Given Trigonometric Ratio’ is an excellent resource prepared by our subject experts for IIT JEE Maths. These effective SL Loney Solutions to all types of questions associated with this chapter will guide you through all the tough problems. In this chapter, you will learn about the general expressions of trigonometric ratios. You know that the equations that involve trigonometric functions of unknown variables are termed as trigonometric equations. Now you also find the solutions in these types of trigonometric equations.

SL Loney Plane Trigonometry General Expressions for All Angles Having a Given Trigonometric Ratio Solutions has a total of 62 questions that have been set-up in two exercises. First is Example XI, it has 29 questions whereas in the second section which is Example XII has 33 questions. All of these questions are based on the General Expressions for All Angles Having a Given Trigonometric Ratios. You have to go through the theorems to find general expressions of trigonometric ratios i.e. sine, cosine, tangent, respectively to tackle the questions of it. Keep practice these questions and do well in your Class 12 board exams as well as entrance exams.

Instasolv is a great resource for preparing trigonometry for JEE. General Expressions for All Angles Having a Given Trigonometric Ratio by SL Loney is one of the significant chapters in trigonometry, and the questions are most frequently asked in competitive exams. That’s why we at Instasolv have provided SL Loney Solutions that will cater all your needs regarding this chapter. We provide you with a step-by-step solution to every question of  SL Loney Plane Trigonometry General Expressions for Trigonometric Ratio. All you need to do is solve questions using our platform you will benefit for sure.

Important Topics of  SL Loney Plane Trigonometry Solutions Chapter 6: General Expressions for All Angles Having a Given Trigonometric Ratio

 In this chapter, You will find three theorems that will help you to determine the general expressions of trigonometric ratios.

You know sine, cosine, and tangent are considered as basic trigonometric ratios. So, we will find general expressions of these ratios. Although the expressions for cosec, secant, and cot can be obtained with the help of the solutions of basic trigonometric ratios. Let’s have a look at them-

Theorem -1: To find the general expressions for all the angles that have the same sine

Let us Consider an equation:

Sin A = Sin B

On solving it, we will find the general expression of it.

So,

Sin A – Sin B = 0

2 cos {(A+B) / 2}. Sin(A-B) / 2 = 0

In this condition,

cos {(A+B) / 2} = 0 or Sin(A-B) / 2 = 0

If we take a common solution from both of these conditions, you will get a general expression.

A = nπ + (-1)nB, where n ∈ Z

Theorem -2: To find the general expressions for all the angles that have the same cosine

Let us consider an expression:

Cos A = Cos B

On solving it, we get –

Cos A – Cos B = 0

2 Sin {(A+ B) / 2} . sin {(B-A) /2} = 0

We can say that –

Sin {(A+ B) / 2} = 0 or sin {(B-A) /2} = 0

{(A+ B) / 2} = nπ or {(B-A) /2} = nπ

These two conditions will get a common general expression-

A = 2nπ± B, where n ∈ Z

Theorem -3: To find the general expressions for all the angles that have same tangent

Similarly consider an equation :

Tan A = Tan B

From this we get –

Sin A / Cos A = Sin B / Cos B

Sin A . Cos B – Cos A . Sin B = 0 { Sin (A – B) = Sin A . Cos B – Cos A . Sin B }

Sin (A – B)=0

From this equation we will get-

(A – B) = nπ

Thus, we will get a general expression –

A = nπ + B, where n ∈ Z.

See here, we have a table that defines the general expressions of trigonometric functions –

Equations Solutions
sin x = 0 x = nπ
cos x = 0 x = (nπ + π/2)
tan x = 0 x = nπ
sin x = 1 x = (2nπ + π/2) = (4n+1)π/2
cos x = 1 x = 2nπ
sin x = sin θ x = nπ + (-1)nθ, where θ ∈ [-π/2, π/2]
cos x = cos θ x = 2nπ ± θ, where θ ∈ (0, π]
tan x = tan θ x = nπ + θ, where θ ∈ (-π/2 , π/2]
sin2 x = sin2 θ x = nπ ± θ
cos2 x = cos2 θ x = nπ ± θ
tan2 x = tan2 θ x = nπ ± θ

Exercise Discussion of SL Loney Plane Trigonometry Solutions Chapter 6: General Expressions for All Angles Having a Given Trigonometric Ratio

Well, we have mentioned earlier that this chapter has two exercises in which questions are settled. We have to solve these questions. Let’s see how it will benefit you-

Exercise XI –

In this exercise, a total of 29 questions are there for you to solve. You will have have to determine the general value of Ѳ, to construct an angle when some trigonometric ratios are given, and in some of them you will be given an expression like this Cos (A-B) = 1/2 and Sin (A+B )= 1/2 at which you need to find the smallest positive values of A and B from it.

Exercise XII –

You will have 33 questions in this exercise. The questions can be quite tough to crack it sometimes. In these questions, you will have trigonometric expressions of unknown variables. All you need to do is find the general value of an expression. Also, you will be asked to find the sin Ѳ, to explain the double result when some trigonometric expression is given.

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