# S.L. Loney Trigonometrical Ratios Multiply and Submultiple Angles Solutions (Chapter 8)

SL Loney Plane Trigonometry Solutions for Chapter 8 ‘Trigonometrical Ratios Multiple and Submultiple Angles’ are created to help you understand the topics of the chapter for competitive exams like IIT JEE Mains and Advanced. This chapter has many solved problems, specimens, and the formula to prove related to trigonometric ratios on angles. It introduces the concepts of trigonometric multiple angles, double angle, triple angle, trigonometric submultiples angle, and half-angle.

SL Loney Solutions Trigonometrical Ratios Multiple and Submultiple Angles Solutions have 87 questions and 3 exercises. SL Loney Plane Trigonometry solutions for JEE provides a good grip on the basic concepts of trigonometrical ratios on multiple and submultiples angles and questions related to them. Problems solved in this book are simple to understand and memorize. The problem-solving technique of this book would help you to solve a question in much less time compared to general methods.

SL Loney Plane Trigonometry Solutions for Trigonometrical Ratios Multiple and Submultiple Angles** **is a useful study material to prepare for competitive exams like NEET and JEE. Our solutions are reliable as they are in sync with the latest syllabus for Class 12 and JEE Maths. Our solutions are checked at various stages to ensure that the answers are 100% accurate and easy to understand.

## Important topics for SL Loney Plane Trigonometry Solutions Chapter 8: Trigonometrical Ratios Of Multiple and Submultiple Angles

There are a few important formulas derived in this chapter. In this section, we will go through a few derivations of formulas and other few important formulas used to solve the problem.

**Trigonometric multiple angles: **sine, cosine, and tangent are general functions for multiple angles. Double angle formula and triple angle formula fall under trigonometric multiple angles. This topic covers derivation for double angle formula and triple angle formula.

**Double angle formula: **If we put A = B in the trigonometric addition and subtraction formula, then we can easily derive the double angle formula. Formulas for sine, cosine, and tangent are as follows:

**sin 2A = 2 sin A cos A**

sin 2A = sin A cos A + cos A sin A

= 2 sin A cos A

**cos 2A = 2 cos**^{2 }A – sin^{2}A

cos 2A = (1- sin^{2} A) – sin^{2} A = 1 – 2 sin^{2} A

cos 2A = 1 – 2 (1 – cos^{2 }A) = 2 cos^{2 }A -1

**tan 2A = 2 tan A/ (1- tan**^{2}A)

**Triple angle formula: **If we put B = 2A in the trigonometric addition and subtraction formula, then we can derive triple angle formula. Formulas for sine and cosine are as follows:

sin 3A = sin (A + 2A)

= sin A cos 2A + cos A sin 2A

= sin A (1 – 2 sin^{2}A) + 2 (1 – sin^{2}A) sin A

= 3 sin A – 4 sin^{3} A

**sin 3A = 3 sin A – 4 sin ^{3} A**

cos 3A = cos (A + 2A) = cos 2A cos A – sin 2A sin A = (2 cos^{2} A – 1) cos A – 2cos A (1 – cos^{2} A)

= 4 cos^{3} A – 3 cos A

**cos 3A = 4 cos ^{3} A – 3 cos A**

**Trigonometric submultiples angles: **Half angle falls under the section of submultiples angles.

**Half angle Formula: **By replacing the A to A/2 in sine, cosine, and tangent double angle formulas we can derive the half-angle formula. Formulas for sine, cosine, and tangent are as follow:

**sin A = 2 sin (A/2) cos (A/2)**

**cos A = 2 cos ^{2 }(A/2) – sin^{2} (A/2)**

**tan A = 2 tan (A/2) / [1- tan ^{2} (A/2)]**

### Exercise Discussion for SL Loney Plane Trigonometry Solutions Chapter 8: Trigonometrical Ratios Of Multiple and Submultiple Angles

SL Loney Trigonometrical Ratios Of Multiple and Submultiple Angles consists of 87 questions that are divided into 3 parts:

- Part 1 of the exercises has 41 solved problems on the topic of multiple angles, double angle, and triple angle. Questions covered in this section are based on finding the value of sine, cosine, and tangent, and proving equations.
- Part 2 of the exercises has 33 solved problems on the topic of submultiples angle and half-angle. Questions covered in this section are based on finding the cosecant, the cosecant of the half-angle, the cosecant of the supplement of double the angle, finding the values of sine, cosine, and tangent, and proving some equations.
- Part 3 of exercises has 13 solved problems based on the entire chapter. Questions covered in this section are based on finding trigonometric ratios, trigonometric functions of an angle, and proving few equations.

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