# S.L. Loney Trigonometrical Ratios The Sum and Difference of Two Angles Solutions (Chapter 7)

SL Loney Plane Trigonometry Solutions Chapter 7 ‘Trigonometrical Ratios Of The Sum and Difference of Two Angles’ is a well-designed solutions book made for those who want to strengthen their conceptual knowledge of Trigonometry for IIT JEE and other engineering entrance exams. In this chapter, you are given several theorems related to the trigonometric functions of sums and differences of two angles in the form of functions of the angles.

SL Loney Trigonometry solutions book for Trigonometrical Ratios Of The Sum and Difference of Two Angles has a total of 72 questions in 4 exercises that will help you revise all the formulas associated with this chapter. You will have questions based on the sum and difference formulas of trigonometric ratios i.e. sine, cosine, and tan respectively. Well, we recommend you just solve these questions as many times as possible, then you will progress in Trigonometrical Ratios Of The Sum and Difference of Two Angles.

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## Important Topics of SL Loney Plane Trigonometry Solutions Chapter 7: Trigonometrical Ratios Of The Sum and Difference of Two Angles

So far we learn trigonometric identities. But, in SL Loney Plane Trigonometry Chapter 7, you will learn several formulas related to trigonometric ratios of addition and subtraction of two angles. Let’s see how to get those formulas using the graphical method-

You can see in this graph shown above, we have a circle (centred at origin O), and which has a radius of 1 unit. We have a point P_{4} on the positive x-axis, having coordinate (1,0), the point P_{1} is situated at an angle ‘x’ from an x-axis, and the coordinate of which is P_{1 }(cos x, sin x). Also, we have a point P_{3} at an angle (x+y) from the x-axis, and the point P_{2} at an angle ‘y’ from x-axis which is measured in the clockwise direction.

In this graph, you will see that Δ OP_{1}P_{3} is congruent to Δ OP_{2}P_{4}. From this, we can deduce that-

➔ P_{1}P_{3}= P_{2}P_{4}

➔ (P_{1}P_{3})^{2}= (P_{2}P_{4})^{2}

We all know the coordinates of these four points.

So, using distance formula we will get –

[cos x – cos (-y)]^{2} + [sin x – sin (-y)]^{2} = [1- cos (x+y)]^{2} + sin^{2}(x+y)

On solving this equation-

**cos(x + y) = cos x cos y – sin x sin y** ………………………………………..(a)

This is the sum formula of two angles of cosine trigonometric function.

Now to get cos(x-y), you have to replace y with (-y) :

So, on replacing you will get –

**cos(x – y) = cos x cos y + sin x sin y**……………………………………..(b)

On replacing, x with π/2, and y with x, in equation (b), you will get-

cos (π/2-x) = sin x……………………………………………………….(c)

And, on replacing x with (π/2-x), in equation (c) you will get-

cos [π/2-( π/2-x)] = sin (π/2-x)

cos x = sin (π/2-x)

sin (π/2-x) = cos x…………………………………………………………(d)

since we see the formula of cos(x+y), I hope you must understand how you could determine the sin(x+y) formula.

Let’s see –

You know sin(x+y) can be written as cos[π/2-(x+y)]

On solving, cos[π/2-(x+y)], you will get-

➔ cos[π/2-(x+y)]

➔ cos [(π/2 – x) – y]

➔ cos (π/2 – x) cos y + sin (π/2 – x) sin y

➔ sin x cos y + cos x sin y

** sin(x+y)= sin x cos y + cos x sin y**……………………………………………..(e)

Similarly, you will find the formula of sin(x-y) by replacing y with -y:

**sin(x-y)= sin x cos y – cos x sin y**………………………………………………(f)

If we substitute the values (π, π/2, 2π) in the equations a, b, e and f. We will have the following relations:

❏ cos (π/2 + x) = -sin x

❏ sin (π/2 + x) = cos x

❏ sin (π – x) = sin x

❏ sin (π + x) = – sin x

❏ cos (π± x) = – cos x

❏ sin (2π – x) = -sin x

❏ cos (2π – x) = cos x

Now, for tan(x+y), we will how we get this-

So, you know tanx = sinx/ cosx

Using it, we can say that-

tan(x+y) = sin(x+y) / cos(x+y))

tan(x+y))= sin x cos y + cos x sin y / cos x cos y – sin x sin y

On dividing numerator and denominator by cosx. cosy, you will get-

**tan(x+y)) = tan x + tan y / 1 – tan x tan y**…………………………………..(g)

Similarly, we will get tan (x – y) on replacing y with -y.

**tan(x+y)) = tan x – tan y / 1 + tan x tan y**……………………………..(h)

** **Apart from this, you will also learn some important formulas that are given below-

❏ sin c + sin D = 2 sin {(C+D) / 2} . cos {(C-D) / 2}

❏ sin c – sin D = 2 cos {(C+D) / 2} . sin {(C-D) / 2}

❏ cos c + cos D = 2 cos {(C+D) / 2} . sin {(C-D) / 2}

❏ cos c – cos D = 2 sin {(C+D) / 2} . sin {(D-C) /2}

You should carefully notice that the second factor in (cos C- cos D ) formula its sin{(D-C)/2} not sin{(C-D)/2}

### Exercise Discussion of SL Loney Plane Trigonometry Solutions Chapter 7: Trigonometrical Ratios Of The Sum and Difference of Two Angles

As mentioned earlier, SL Loney Solutions include 72 questions in 4 exercises. You will get to know all types of sum and differences formulas after solving them.

**Exercise I **

The first set of questions is based on the formulas given in above equations a, b, e, f respectively. You will be given 3 examples and 12 solved problems. Solving them once, you will be able to cope with questions based on it. Questions are generally proving based and in some of them, you need to represent the answer graphically.

**Exercise II **

In this set of questions, you will be given the trigonometric expressions. All you need to do is simplify the expressions. It has around 31 questions with 4 useful examples. Questions are based on the important formulas mentioned above. These questions are quite interesting and can be tricky sometimes to solve it.

**Exercise III **

You will have 17 odd questions with 4 examples as well based on the formulas given in above equations a, b, e, and f respectively. However, in these questions, you need to express the trigonometric ratios as a sum or difference of two angles. You will also have questions on the version (A+B), and version (A-B), etc.

**Exercise IV**

The last set of questions based on the formulas given in equation g and h. You will be asked to solve 14 questions in which 2 are examples. Practising it once you will be able to tackle the questions of this section.

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