# S.L. Loney Relations Between The Side and Angles Solutions (Chapter 12)

SL Loney Plane Trigonometry Solutions for Chapter 12 ‘Relations between the Sides of a Triangle and its Angles’ include exercise solutions of the chapter created by the best trigonometry experts at Instasolv. These solutions describe in detail how the angles in any triangle are related to their corresponding sides by means of trigonometric functions. These solutions will be useful for the preparation of competitive exams like IIT JEE and NEET as questions related to relations between sides of a triangle and its angle are often asked in these exams.

SL Loney Plane Trigonometry Solutions for JEE Relations between the Sides of a Triangle and its Angles have a total of 46 questions which are divided into 2 exercises. In these exercises, you will be able to learn the relationship between how the sides of a triangle and different trigonometrical functions like Sine, Cosine, and Tangent are related to each other. You also learn the same relations when the angels are halved and so you would get to see the same idea in different forms. Having complete Solutions to SL Loney questions is important to form a strong base in trigonometry. Our solutions will definitely help you in solving the exercise problems which would help you bypassing conceptual hurdles.

Getting solutions to difficult exercise questions from experts is a good way of learning and understanding the concepts since experts will provide a deep insight into the concepts involved. Our team at Instasolv provides complete solutions to SL Loney Plane Trigonometry Part 1 to better equip you with the kind of questions that come in major competitive exams like IIT-JEE and NEET along with Class 12 syllabus. With our solutions, you also learn how to do questions efficiently and ensure better time-management during stressful exam environments.

## Important Topics for SL Loney Plane Trigonometry Solutions Chapter 12: Relations Between The Sides of a Triangle and Its Angles

**Relation Between sides and Sines of the Angles Opposite to Them – **In any triangle, say ABC, the side opposite to angle A which is BC is represented by a, similarly for angles B the side is AC represented by b and for angle C it is AB represented by c. This is shown in the figure below:

In the figure above, if line AD is drawn perpendicular to BC then :

In triangle ABD, Sin B = AD/AB or AD = c Sin B – (i)

In triangle ACD, Sin C = AD/AC or AD = b Sin C – (ii)

From (i) and (ii) we get c Sin B = b Sin C, or c/Sin C = b/Sin B

In a similar way, a perpendicular can be drawn from B on CA from which we will get:

c/Sin C = a/Sin A

Combining them all we get a/Sin A = b/Sin B = c/ Sin C

**Cosine of an Angle in Terms of its Sides – **From figure 1, as per Euclid’s theorem:

AB^{2} = BC^{2} + CA^{2} – 2 BC CD – (i)

Cos C = CD/CA hence CD = CA Cos C = b Cos C

So (i) can be written as: c^{2} = a^{2} + b^{2} – 2 a b Cos C

Hence Cos C = a^{2} + b^{2 }– c^{2}/2 a b

Similarly it can be proved that Cos B = c^{2} + a^{2 }– b^{2}/2 a c

And Cos A = b^{2} + c^{2 }– a^{2}/2 b c

**Sines of Half The Angles in Terms of its Sides – **From the above derivation, we can combine another formula which says:

Cos A = 1 – 2 Sin^{2} A/2

We get 2 Sin^{2} A/2 = 1 – Cos A = 1 – ( b^{2} + c^{2 }– a^{2}/2 b c)

This can be broken down to express it as:

Sin A/2 = √((s- b) (s – c))/ bc

Sin B/2 = √((s- a) (s – c))/ ac

Sin C/2 = √((s- a) (s – b))/ ab

Here s = (a + b + c)/2

**Cosines of Half the Angles in Terms of its Sides – **We know that

Cos A = 2 Cos^{2} A/2 – 1

Hence 2 Cos^{2} A/2 = 1 + Cos A = 1 + (b^{2} + c^{2 }– a^{2})/2 b c

Finally we get : Cos A/2 = √(s (s – a))/bc

Cos B/2 = √(s * (s – b))/ac

Cos C/2 = √(s * (s – c))/ab

**Tangents of Half the Angles in Terms of its Sides – **We know that

Tan A/2 = (Sin A/2)/ Cos A/2

So substituting from above derivations we get:

Tan A/2 =** (**√((s- b) (s – c))/ bc ) / √(s * (s – a))/bc

= √((s- b) (s – c))/ s * (s – a)

Similarly Tan B/2 = √((s- a) (s – c))/ s * (s – b)

Tan C/2 = √((s- a) (s – b))/ s * (s – c)

Since in a triangle any angle is always < 180º, hence A/2 is always < 90º which means sine, cosine and tangent of A/2 is always positive.

Few other important formulas relating sides of a triangle and the angles are:

- a = b Cos C + c Cos B, b = a Cos C + c Cos A, a = a Cos B + b Cos C
- If sides of a triangle are in an arithmetic progression then its cotangents of half angles are also in arithmetic progression. So, if a + c = 2 b, then Cot A/2 + Cot C/2 = Cot B/2

### Discussion of Exercises of SL Loney Plane Trigonometry Solutions Chapter 12: Relations Between The Sides of a Triangle and Its Angles

- The first part of the exercises for Relations between the sides of a Triangle and its Angles by SL Loney Solutions has 7 questions.
- In these questions, you are given values for sides of a triangle and you need to find Sin, Cos, and tan of corresponding angles and half angles. In one question, you also need to verify it with a graph.
- In the second set of exercises, there are 39 problems where you need to prove many trigonometrical formulas for angles and sides of a triangle based on the theorems discussed.
- Some theorem derivations are also included in this exercise. Questions also involve the right-angle and isosceles triangles and the relations between their sides and angles.

## Why Use SL Loney Plane Trigonometry Solutions Chapter 12: Relations between the Sides of a Triangle and its Angles by Instasolv?

- Relations between the Sides of a Triangle and its Angles by SL Loney Plane Trigonometry Part 1 involves many important concepts of triangles and trigonometrical relations.
- You also get to solve the kind of questions that can come in the JEE or Mathematics Olympiad kind of competitions.
- You must seek solutions to Plane Trigonometry by SL Loney from our experts which will clarify your doubts and strengthen your base
**.** - Our team of experts always strives to give you the best approach for all the problems.
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