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# SL Loney Trigonometry Solutions

SL Loney Trigonometry Solutions are the best material for the comprehensive practice of the topic ‘Trigonometry’. S.L. Loney Plane Trigonometry book is divided into two parts where part-1 contains the syllabus you need to follow for appearing into class 12 board examination, and part-2 helps you prepare well for competitive exams. Our solutions are a complete treatise on this topic, providing a holistic knowledge of all trigonometry concepts. Trigonometry is an important topic from the examination point of view.

SL Loney Solutions for Trigonometry is recommended by all the toppers and mathematics faculties. Our Solutions have been prepared by a team of experts for your easy reference and effective learning. You can practise questions based on topics such as measurements of angles, trigonometric ratios of angles less than a right angle, heights and distances, applications of algebraic signs to trigonometric ratios, general expressions of all angles having a given trigonometric ratio, ratios of the sum and difference of two angles, ratios of multiple and submultiple angles, identities and trigonometric equations and logarithms. Ample amount of questions of all levels are given in the chapters, and we suggest you refer to our solutions for a stepwise method for solving these questions.

SL Loney Plane Trigonometry solutions have been carefully curated using the best methods for solving trigonometry-based questions. These help you in building your concepts in the early stages of learning and act as a ready reckoner during exam preparation and revision. You can surely attempt questions based on trigonometry and score good marks in all the exams with these solutions.

S.L. Loney

## SL Loney Plane Trigonometry Solutions: Chapter-wise

S.L. Loney Trigonometry Book comprises 20 chapters. The constituents of relevant chapters of the book are discussed below.

## SL Loney Trigonometry Chapter (Measurements of Angles, Sexagesimal, Centesimal and Circular Measure) Solutions

In this chapter, you learn about the sexagesimal system where the right angle is divided into 90 equal parts called degrees and in the centesimal system right angles are divided into 100 equal parts. You also learn to interchange between these two systems. Next, you learn about the third system called circular measures of angles. Here you learn about radians, the magnitude of radians and some theorems. There is a total of 86 questions spread through three exercises.

## SL Loney Plane Trigonometry Chapter 2 (Trigonometrical ratios For Angles Less Than A Right Angle) Solutions

Chapter 2 of SL Loney Trigonometry Solutions, you learn about the various trigonometric ratios with names sine, cosine, tangent, cotangent, cosecant and secant, of a right-angled triangle. Here you learn that trigonometrical ratios are always the same for the same angle. Next, you learn about the fundamental relations between the trigonometrical ratios of an angle in terms of equations and complementary angles. This chapter consists of 66 questions in three exercises.

## SL Loney Trigonometry Chapter 3 (Simple Problems in Heights and Distances) Solutions

The third chapter of S.L. Loney Plane Trigonometry solutions deals with the most important objective of trigonometry, which is finding distances between points, or the heights of objects without actually measuring these distances or heights. You learn about angles of elevation and depression here. This chapter has 21 word problems.

## SL Loney Plane Trigonometry Chapter 4 (Application of Algebraic Signs to Trigonometry) Solutions

In chapter 4, you learn about positive and negative angles in terms of clockwise and counterclockwise movements and positive and negative lines. Next, you learn about, trigonometrical ratios for an angle of any magnitude, Versed sine and Coversed sine concepts. Then you learn about some relations of trigonometric ratios and signs of the trigonometrical ratios in all the four quadrants.

This chapter also teaches you about tracing the changes in the sign and magnitude of the trigonometrical ratios of an angle, as the angle increases from 0° to 360°. Lastly, this chapter tells you about periods of the trigonometrical functions and sine, cosine, tangent, cotangent, cosecant and secant curves. There are 12 miscellaneous questions in this chapter.

## SL Loney Trigonometry Chapter 5 (Trigonometrical Functions of Angles of any Size and Sign) Solutions

The fifth chapter of SL Loney Trigonometry book teaches you about how to find the trigonometric ratios of an angle (-θ) and θ, (90° – θ) in terms of those of θ, for all values of θ. Next, you learn how to find the trigonometrical ratios of the angle (90° + θ), (180° – θ), (360° + θ). You also learn about supplementary angles here. This chapter contains 38 questions, some of which ask you to prove equalities in trigonometric equations.

## SL Loney Plane Trigonometry Chapter 6 (General Expressions for All Angles Having a Given Trigonometrical Ratio) Solutions

This chapter teaches you to construct the least positive angle whose sine is equal to a, where a is a proper function and similar statements for other trigonometric functions. Next, you learn some theorems regarding some general expression to include all angles having the same sine, cosine and tangent. Further, you learn about trigonometrical equations and methods to solve them for all angles. This chapter has two exercises containing a total of 62 questions.

## SL Loney Trigonometry Chapter 7 (Trigonometrical Ratios of the Sum and Differences of two Angles) Solutions

This chapter of SL Loney Trigonometry solutions teaches you some theorems based on proving some trigonometrical equations like sin (A + B) = sin A cos B + cos A sin B and cos (A + B) = cos A cos B – sin A sin B and some more related theorems. These theorems are true for all angles, which are not greater than two right angles. This chapter has 62 questions spread through three exercises.

## SL Loney Plane Trigonometry Chapter 8 (The Trigonometrical Ratios of Multiple and Submultiple Angles) Solutions

Here you learn to find the trigonometrical ratios of angles 2A and 3A in terms of those of the angle A. Next, you will learn to express the trigonometrical ratios of any higher multiples of θ. Further, this chapter teaches you about submultiple angles like A/2 in terms of sin A, cos A, and tan A. This chapter further teaches you to use the above formulas to find the trigonometrical ratios of some important angles like 18°, 36°, 9°. This chapter has 87 questions which let you practise all the concepts taught in the chapter.

## SL Loney Trigonometry Chapter 9 (Identities and Trigonometrical Equations) Solutions

The 9th chapter of S.L. Loney Plane Trigonometry solutions teaches you that the formulas learnt in earlier chapters can be used to obtain the trigonometrical ratios of the sum of more than two angles. Next, you learn about a theorem which gives the tangent of the sum of any number of angles in terms of the tangents of the angles themselves. Further, this chapter teaches you about the identities holding between the trigonometrical ratios of the angles of a triangle, when three angles A, B and C are such that their sum is 180°.

This chapter also teaches you to use addition and subtraction theorems to solve trigonometrical equations and use of subsidiary angles to facilitate computation. This chapter has 83 questions based on the above learnings.

## SL Loney Plane Trigonometry Chapter 10 (Logarithms) Solutions

This chapter introduces you to the concept of logarithms which says if a be any number and x and N two other numbers such that ax = N, then x is called the logarithm of N to the base a and is written logaN. Further, you learn about the three fundamental laws of logarithms – (i) loga(mn) = logam + logan; (ii) loga(m/n) = logam – logan; (iii) loga mn = n loga m. Further in this chapter, you learn about the common system of logarithms, characteristics of Mantissa, characteristics of logarithms of decimal numbers. Next, you get to know all about tables of logarithms, also known as Chamber’s tables. This chapter has 12 questions to solve.

## SL Loney Trigonometry Chapter 11 (Tables of Logarithms and Trigonometrical Ratios, Principle of Proportional Parts) Solutions

In this chapter of our SL Loney Plane Trigonometry Solutions, you learn about the principle of proportional parts which states that the increase in the logarithm of a number is proportional to the increase in the number itself and it is true with certain limitations. Next, you learn advance usage of the table of trigonometrical ratios where you can use an inverse equation to find an angle. You also learn about tables of logarithmic sines and cosines etc. There are 28 total questions in two exercises of this chapter.

## SL Loney Plane Trigonometry Chapter 12 (Relations Between the Sides and the Trigonometrical Ratios of the Angles of any Triangles) Solutions

This chapter teaches you that in any triangles ABC, the side BC, opposite to the angle A, is denoted by a; the sides CA and AB, opposite to the angles B and C respectively, are denoted by b and c, and sin A/a = sin B/b = sin C/c, i.e. the sines of the angles are proportional to the opposite sides. This information is used to create and prove many identities and equations. This chapter contains 40 questions based on the above concepts.

## SL Loney Trigonometry Chapter 13 (Solution of Triangles) Solutions

The chapter number 13 of SL Loney Plane Trigonometry Solutions teach you that in any triangle the 3 sides and the 3 angles are called elements of the triangle, and when any three elements are given, we can get information about the triangle. When 3 elements of a triangle are given, the process of calculating its other 3 elements is called the solution pf the triangle. You also learn about some geometrically ambiguous cases of the solutions of triangles. This chapter has 64 questions in 5 exercises.

## SL Loney Plane Trigonometry Chapter 14 (Heights and Distances) Solutions

This chapter of SL Loney Trigonometry Book teaches you the methods of finding the height of an inaccessible tower by means of observations made a distant point, to find the distances between two inaccessible points by means of observations made at two points, and the distance between which is known, where all the four points are in a single plane. You also learn about bearings and points of the compass. There are a total of 63 questions of all levels from these concepts in 2 exercises.

## SL Loney Trigonometry Chapter 15 (Properties of a Triangle) Solutions

This chapter teaches you to calculate the area of a triangle in terms of trigonometric ratios where, for a triangle ABC, Δ = 1/2ca sin B = 1/2ab sin C = 1/2bc sin A. Next you learn about a circle which passes through the angular points of a triangle ABC and is known as circumcircle. If the circle is inside a triangle, it is called incircle. Here you learn to calculate various parameters of such triangles with the help of parameters of circles. You also learn about orthocenter and pedal triangle of any triangle; centroid and median of any triangle; bisectors of the angles in a triangle and finding their sides and angles. This chapter contains a total of 79 questions in 3 exercises.

## SL Loney Plane Trigonometry Chapter 16 (On Quadrilaterals and Regular Polygons) Solutions

In this chapter of S.L. Loney Plane Trigonometry book, you learn to find the area of a quadrilateral which is inscribable in a circle; regular polygons and radii of the inscribed and circumscribing circles of a regular polygon; and area of a regular polygon. There are 32 questions in 2 exercises of this chapter.

## SL Loney Trigonometry Chapter 17 (Trigonometrical Ratios of Small Angles, Area of a Circle, Dip of the Horizon) Solutions

This chapter teaches you that if θ be the number of radians in any angle, which is less than a right angle, then sin θ, θ and tan θ are in ascending order of magnitude. You also learn to determine the area of a circle in terms of a regular polygon, and area of the sector of a circle. Next, you learn about dip of the horizon, offing or visible horizon and formulas related to these. There are altogether 22 questions in 3 different exercises of this chapter.

## SL Loney Plane Trigonometry Chapter 18 (Inverse Circular Functions) Solutions

In chapter number 18 of our SL Loney Plane Trigonometry, you will learn about sin-1 a which is being read as ‘sine minus one a’ and is different from 1/sin a. This is an angle which denotes the smallest numerical angle whose sine is a. Similarly, you learn about cos-1, sec-1, cot-1 etc. All of these quantities are known as inverse circular functions. This chapter contains 44 questions in 1 exercise.

## SL Loney Trigonometry Chapter 19 (On Some Simple Trigonometrical Series) Solutions

In this chapter, you learn to find the sum of the sines and cosines of a series of angles, the angles being in arithmetical progression. You come to know about various expressions related to these. There are 27 questions in this chapter.

## SL Loney Plane Trigonometry Chapter 20 (Elimination) Solutions

This chapter teaches you that when you have two equations each containing one unknown quantity, and there is a relation between the constants of both equations so that the same value of the unknown quantity (say x) may satisfy both, so by eliminating the quantity x we can reach to a simplified equation for solving the equations. Here you learn the effective methods of solving equations by elimination. This chapter contains 13 questions in one exercise of SL Loney Plane Trigonometry solutions.