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# RS Aggarwal Solutions Class 10 Chapter 8 - Trigonometric Identities (Ex 8A) Exercise 8.1

Last updated date: 17th Sep 2024
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## RS Aggarwal Solutions Class 10 Chapter 8 - Trigonometric Identities (Ex 8A) Exercise 8.1 - Free PDF

Competitive Exams after 12th Science

## Chapter 8 - Trigonometric Identities

### Trigonometric Identities

Equalities involving trigonometric functions are known as Trigonometric Identities, and they hold for all values of the variables involved, defining both sides of the equality. We'll look at Trigonometric Identities in this mini-lesson. The three primary trigonometric ratios are sin, cos, and tan. The reciprocals of sin, cos, and tan are the three other trigonometric ratios in trigonometry: sec, cosec, and cot. What is the relationship between these trigonometric ratios (sin, cos, tan, sec, cosec, and cot)? Trigonometric Identities bind them together (or in short trig identities). In the following sections, we'll look at the Trigonometric Identities in greater depth.

### What exactly are Trigonometric Identities?

Trigonometric Identities are equations that are true for any value of the variable in the domain and relate to various trigonometric functions. A mathematical expression that holds for all values of the variable(s) it contains is called an identity.

For instance, here are some algebraic identities:

(m + n)2 = m2 + 2mn + n2

(m - n)2 = m2 - 2mn + n2

(m + n)(m - n)= m2 - n2

The variables are only addressed in the algebraic identities, whereas the six trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent are addressed in the trig identities. Let's take a closer look at each type of trigonometric identity.

### Reciprocal Trigonometric Identities

We already know that cosecant, secant, and cotangent are the reciprocals of sin, cosine, and tangent, respectively.

As a result, the reciprocal identities are as follows:

• sin θ = 1/cosecθ (OR) cosec θ = 1/sinθ

• cos θ = 1/secθ (OR) sec θ = 1/cosθ

• tan θ = 1/cotθ (OR) cot θ = 1/tanθ

### Important Trigonometric Identities Notes

• We use the pairs (sin, cos), (cosec, sec), and (tan, cot) to write the trigonometric ratios of complementary angles.

• The trigonometric ratio will not change while writing the trigonometric ratios of supplementary angles. The fact that only sin and cosec are positive in the second quadrant where the angle is of the form (180-θ) can be used to determine the sign.

• The cos 2x formula has three different formulas. Only the first one is worth remembering because the other two can be found using the Pythagorean identity sin2x + cos2x = 1.

• Applying the identity tan = sin/cos and then using the half-angle formulas of sin and cos yields the tan half-angle formula.

## FAQs on RS Aggarwal Solutions Class 10 Chapter 8 - Trigonometric Identities (Ex 8A) Exercise 8.1

1. What are Pythagorean Trigonometric Identities according to RS Aggarwal Solutions Class 10 Chapter 8 - Trigonometric Identities (Ex 8A) Exercise 8?

The Pythagorean Trigonometric Identities are derived from Pythagoras' theorem in trigonometry. The Pythagoras theorem is applied to the right-angled triangle below, yielding:

Dividing both sides by Hypotenuse2

sin2θ + cos2θ = 1

One of the Pythagorean identities is this. We can derive two more Pythagorean Trigonometric Identities in the same way.

• 1 + tan2θ = sec2θ

• 1 + cot2θ = cosec2θ

Now you might have a clear understanding of what Pythagorean Trigonometric Identities mean and also their kinds.

2. What do you contemplate with complementary and supplementary Trigonometric Identities according to RS Aggarwal Solutions Class 10 Chapter 8 - Trigonometric Identities (Ex 8A) Exercise 8?

Complementary angles are a pair of two angles whose sum equals 90°.

• sin (90°- θ) = cos θ

• cos (90°- θ) = sin θ

• cosec (90°- θ) = sec θ

• sec (90°- θ) = cosec θ

• tan (90°- θ) = cot θ

• cot (90°- θ) = tan θ

The supplementary angles are a pair of two angles whose sum equals 180°.

• sin (180°- θ) = sinθ

• cos (180°- θ) = -cos θ

• cosec (180°- θ) = cosec θ

• sec (180°- θ)= -sec θ

• tan (180°- θ) = -tan θ

• cot (180°- θ) = -cot θ

3. What do you understand by sum and difference Trigonometric Identities?

The formulas sin(X+Y), cos(X-Y), cot(X+Y), and others are part of the sum and difference identities.

• sin (X+Y) = sin X cos Y + cos X sin Y

• sin (X-Y) = sin X cos Y - cos X sin Y

• cos (X+Y) = cos X cos Y - sin X sin Y

• cos (X-Y) = cos X cos Y + sin X sin Y

• $\tan \left ( X+Y \right ) = \frac{\left ( \tan X+\tan Y \right )}{1-\tan X \tan Y }$

• $\tan \left ( X-Y \right ) = \frac{\left ( \tan X-\tan Y \right )}{1+\tan X \tan Y }$

4. How is a Sine Rule described in RS Aggarwal Solutions Class 10 Chapter 8 - Trigonometric Identities (Ex 8A) Exercise 8?

It is the ratio of the side opposite a given angle (in a right-angled triangle) to the hypotenuse is the trigonometric function. The sine rule describes the relationship between a triangle's angles and their corresponding sides. We'll have to use the sine rule to solve non-right-angled triangles. The sine rule can be written as follows for a triangle with sides 'x', 'y', and 'z' and opposite angles X, Y, and Z.

• $\frac{x}{\sin X}=\frac{y}{\sin Y}=\frac{z}{\sin Z}$

• $\frac{\sin X}{x}=\frac{\sin Y}{y}=\frac{\sin Z}{z}$

• $\frac{x}{y}=\frac{\sin X}{\sin Y};\frac{y}{z}=\frac{\sin Y }{\sin Z};\frac{x}{z}=\frac{\sin X}{\sin Z}$

5. What is a Cosine Rule according to RS Aggarwal Solutions Class 10 Chapter 8 - Trigonometric Identities (Ex 8A) Exercise 8?

It is the ratio of the side adjacent to an acute angle (in a right-angled triangle) to the hypotenuse is the trigonometric function. When two sides and the included angle of a triangle are given, the cosine rule is used to determine the relationship between the angles and the sides. The sine rule can be written as follows for a triangle with sides 'x', 'y', and 'z' and opposite angles X, Y, and Z.

• x2 = y2 + z2 - 2yz·cosX

• y2 = z2 + x2 - 2zx·cosY

• z2 = x2 + y2 - 2xy·cosZ