This GSEB Class 10 Maths Notes Chapter 8 Introduction to Trigonometry covers all the important topics and concepts as mentioned in the chapter.

## Introduction to Trigonometry Class 10 GSEB Notes

1. If a student is looking at the top of the Minar, a right angled can be imagined to be made, as shown below.

2. Suppose a girl is sitting on the balcony of her house located on the bank of a river. She is looking down at a flower pot placed on a stair of a temple situated nearby on the other bank of a river. A right triangle is made in this situation as shown below.

In all the situations given above, the distances or heights can be found by using some mathematical techniques, which come under a branch of mathematics called â€˜trigonometryâ€™. The word â€˜trigonometryâ€™ is derived from the Greek words â€˜triâ€™ (meaning three) â€˜gonâ€™ (meaning sides), and â€˜metronâ€™ (meaning measure). In fact, trigonometry is the study of relationships between the sides and angles of a triangle.

In this Chapter, we will study some ratios of the sides of a right triangle with respect to its acute angles called trigonometric ratios of the angle. We will also define the trigonometric ratios for angles of measure 0Â° and 90Â°. We will calculate trigonometric ratios for some specific angles and establish some identities involving these ratios called trigonometric identities.

**Trigonometric Ratios**

Let us take a right triangle ABC as shown in the fig.

Here, âˆ CAB (or, in brief, angle A) is an acute angle. Note the position of the side BC with respect to angle A. It faces âˆ A. We call it the side opposite to angle A. AC is the hypotenuse of the right triangle and the side AB is adjacent to âˆ A. So, we call it the side adjacent to angle A.

Note that the position of sides change when you consider angle C in place of A (see fig.)

Aryabhatta A.D. 476-550

retained as it is. The word jiva was translated into sinus, which means curve, when the Arabic version was translated into Latin. Soon the word sinus also used as sine, became common in mathematical texts throughout Europe. An English Professor of astronomy Edmund Gunter (1581 – 1626), first used the abbreviated notation â€˜sinâ€™.

The origin of the terms â€˜cosineâ€™ and â€˜tangentâ€™ is much later. The cosine function arose from the need to compute the sine of the complementary angle. Aryabhatta called it kotijya. The name cosinus originated with Edmund Gunter. In 1674, the English Mathematician Sir. Jonas Moore first used the abbreviated notation â€˜cosâ€™.

Remark: Symbol sin A is used as an abbreviation for â€˜sine of angle Aâ€™. It is not the product of â€˜sin and Aâ€™.

Similar interpretations follow for other trigonometric ratios also.

- sine A is abbreviated as sin A
- cosine A is abbreviated as cos A
- tangent A is abbreviated as tan A
- cotangent A is abbreviated as cot A
- secant A is abbreviated as sec A
- cosecant A is abbreviated as cosec A.

**Trigonometric Ratios of a given Acute Angie :**

Let âˆ QAN be any acute angle, where AQ and AN are two arms of an angle.

Draw PM perpendicular to AN and CB perpendicular to AN.

In right angled triangle APN,

Similarly, in right angled âˆ†ABC, CB

In âˆ†APM and âˆ†ACB,

âˆ AMP = âˆ ABC

âˆ CAB = âˆ CAB

âˆ´ âˆ†APM ~ âˆ†ACB [By AA similarity]

[From (1) and (2)]

This shows that the trigonometric ratios of angle A in âˆ†PAM not differ from those of angle A in âˆ†CAB.

So, we conclude that the values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle if angle remains saine.

**Trigonometric Ratios of Some Specific Angles**

In this topic, we will find Trigonometric Ratio of Angles : 0Â°, 30Â°. 45Â°, 60Â° and 90Â°.

**Trigonometric Ratios of 0Â° and 90Â°**

Let us see what happens to the trigonometric ratios of angle A, if it is made smaller and smaller in the right triangle ABC (see fig.), till it becomes zero.

As âˆ A gets smaller and smaller, the length of the side BC decreases. The point C gets closer to point B, and finally when âˆ A becomes very close to 0Â°, AC becomes almost the same as AB (see fig.)

When âˆ A is very close to 0, BC gets very close to 0 and so the value of sin A = \(\frac{B C}{A C}\) is very close to 0. Also, when âˆ A is very close to 0, AC is nearly the same as AB and so the value of cos A = \(\frac{A B}{A C}\)

This helps us to see how we can define the values of sin A and cos A when A = 0Â°. We define :

sin 0Â° = 0 and cos 0Â° = 1.

Using these, we have :

tan0Â° = \(\frac{\sin 0^{\circ}}{\cos 0^{\circ}}\) = 0

cot0Â° = \(\frac{1}{\tan 0^{\circ}}\)

which is not defined (Why) ?

sec 0Â° = \(\frac{1}{\cos \theta^{\circ}}\) = 1 and cosec 0Â° = \(\frac{1}{\sin 0^{\circ}}\)

which is again not defined. (Why) ?

Remark:

cosec 0Â° = \(\frac{1}{\sin 0^{\circ}}=\frac{1}{0}\) = not defined

cot 0Â° = \(\frac{1}{\tan 0^{\circ}}=\frac{1}{0}\) = not defined.

Now, let us see what happens to the trigonometric ratios of âˆ A when it is made larger and larger in âˆ†ABC till it happens 90Â°. As âˆ A gets larger and larger, âˆ C gets smaller and smaller. Therefore, as in the case above, the length of the side AB goes on decreasing. The point A gets closer to point B. Finally when âˆ A is very close to 90Â°, âˆ C becomes very close to 0Â° and the side AC almost coincides with side BC (see fig).

When âˆ C is very close to 0Â°, âˆ A is very close to 90Â°, side AC is nearly the same as side BC, and so sin A is very close to 1. Also when âˆ A is very close to 90Â°, âˆ C is very close to 0Â°, and the side AB is nearly zero, so cos A is very close to 0.

So, we define : sin 90Â° = 1 and cos 90Â° = 0.

Remark :

- tan 90 = \(\frac{\sin 90^{\circ}}{\cos 90^{\circ}}=\frac{1}{0}\) = not defined
- sec 90Â° = \(\frac{1}{\cos 90^{\circ}}=\frac{1}{0}\) = not defined .

**Trigonometric Ratios of 30Â° and 60Â°**

Let us now calculate the trigonometric ratios of 30Â° and 60Â°. Consider an equilateral triangle ABC. Since each angle in an equilateral triangle is 60Â°, therefore,

âˆ A = âˆ B = âˆ C = 60Â°

Draw a perpendicular AD from A to the side BC (see fig.)

Now âˆ†ABD = âˆ† ACD

Therefore, BD = DC

and âˆ BAD = âˆ CAD (CPCT)

Now observe that:

âˆ†ABD is a right triangle, right angled at D with âˆ BAD = 30Â° and âˆ ABD = 60Â° (see fig.).

As you know, for finding the trigonometric ratios, we need to know the length of the sides of the triangle. So let us suppose that AB = 2a.

Then BD = \(\frac{1}{2}\)BC = a

and AD^{2} = AB^{2} – (BD)^{2}

= (2a)^{2} – (a)^{2} = 3a^{2},

Therefore, AD = a\(\sqrt{3}\)

Now, we have:

Similarly,

- sin 60Â° = \(\frac{\mathrm{AD}}{\mathrm{AB}}=\frac{a \sqrt{3}}{2 a}=\frac{\sqrt{3}}{2}\)
- cos 60Â° = \(\frac{1}{2}\)
- tan 60Â° = \(\sqrt{2}\)
- cosec 60Â° = \(\frac{2}{\sqrt{3}}\)
- sec 60Â° = 2
- cot 60Â° = \(\frac{1}{\sqrt{3}}\)

**Trigonometric Ratios of 45Â°:**

In âˆ†ABC, right-angled at B, if one angle is 45Â°, then the other angle is 45Â°,

i.e., âˆ A = âˆ C = 45Â° (see fig.)

So, BC=AB

[Equal angles have equal sides opposite to it]

Now suppose BC = AB = a

Then by Pythagoras Theorem,

AC^{2} = AB^{2 }+ BC^{2}

= a^{2} + a^{2 }= 2,

and therefore, AC = a\(\sqrt{2}\)

To find T-ratios of âˆ A = 45Â°

Here Hyp. AC = a\(\sqrt{2}\)

Base (AB) = a

Perpendicular (BC) = a

Using the definitions of the trigonometric ratios, we have:

Table for Recalling different values of T-ratios.

**Trigonometric Ratios of Complementary Angles**

We know that two angles are said to be complementary if sum of two angles is equal to 90Â°. Consider a right angled at B.

Since âˆ A + âˆ C = 90Â°.

To find T-ratios of angle A.

Hyp. = AC

Base = AB

Perpendicular = BC

Now we will find out the T-ratios of angle

90Â° – A = âˆ C

For âˆ C = 90Â° – A

Hyp. = AC

Base = BC

Perp. = AB

Compare the ratios in (1) and (2) AB

- sin (90Â° – A) = \(\frac{\mathrm{AB}}{\mathrm{AC}}\) = cos A
- cos (90Â° – A) = \(\frac{\mathrm{BC}}{\mathrm{AC}}\) = sin A
- tan (90Â° – A) = \(\frac{\mathrm{AB}}{\mathrm{BC}}\) = cot A
- cot (90Â° – A) = \(\frac{\mathrm{BC}}{\mathrm{AB}}\) = tan A
- cosec (90Â° – A) = \(\frac{\mathrm{AC}}{\mathrm{AB}}\) = sec A
- sec (90Â° – A) = \(\frac{\mathrm{AC}}{\mathrm{BC}}\) = cosec A

Remarks: When we will change the angle then sin Î¸, cos Î¸, tan Î¸. cot Î¸, sec Î¸, cosec Î¸ will also change. To remember the changes add â€˜coâ€™ if it is not added, remove â€˜coâ€™ if it is added.

**Trigonometric Identities**

An equation is called an identity when it is true for the values of the variables involved.

Similarly, an equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true to all values of the angle(s) involved.

In this section, we will prove the trigonometric identity, and use it further to prove other useful trigonometric identities.

In âˆ†ABC, right angled at B (see fig.), we have :

AB^{2} + BC^{2} = AC^{2} …(1)

Dividing each term (1) by AC^{2}, we get:

\(\frac{\mathrm{AB}^{2}}{\mathrm{AC}^{2}}+\frac{\mathrm{BC}^{2}}{\mathrm{AC}^{2}}=\frac{\mathrm{AC}^{2}}{\mathrm{AC}^{2}}\)

i.e., \(\left(\frac{\mathrm{AB}}{\mathrm{AC}}\right)^{2}+\left(\frac{\mathrm{BC}}{\mathrm{AC}}\right)^{2}=\left(\frac{\mathrm{AC}}{\mathrm{AC}}\right)^{2}\)

i.e., (cos A)^{2} + (sin A)^{2} = 1

i.e., cos^{2} A + sin^{2} A = 1. …………(2)

This is true for all A such that 0Â° â‰¤ A â‰¤ 90Â°

So, this is a trigonometric identity.

Let us now divide (1) by AB^{2}, we get:

\(\frac{\mathrm{AB}^{2}}{\mathrm{AB}^{2}}+\frac{\mathrm{BC}^{2}}{\mathrm{AB}^{2}}=\frac{\mathrm{AC}^{2}}{\mathrm{AB}^{2}}\)

or

\(\left(\frac{\mathrm{AB}}{\mathrm{AB}}\right)^{2}+\left(\frac{\mathrm{BC}}{\mathrm{AB}}\right)^{2}=\left(\frac{\mathrm{AC}}{\mathrm{AB}}\right)^{2}\)

i.e., 1 + tan^{2}A = sec^{2} A …(3)

Let us see what we get on dividing (1) by BC^{2}. we get:

\(\frac{\mathrm{AB}^{2}}{\mathrm{BC}^{2}}+\frac{\mathrm{BC}^{2}}{\mathrm{BC}^{2}}=\frac{\mathrm{AC}^{2}}{\mathrm{BC}^{2}}\)

i.e \(\left(\frac{\mathrm{AB}}{\mathrm{BC}}\right)^{2}+\left(\frac{\mathrm{BC}}{\mathrm{BC}}\right)^{2}=\left(\frac{\mathrm{AC}}{\mathrm{BC}}\right)^{2}\)

i.e., cot^{2}A + 1 = cosec^{2} A …(4)

Using these identities, we can express each trigonometric ratio in terms of other trigonometric ratios.

i.e., if any one of the ratios is known, we can also determine the values of other trigonometric ratios.