Introduction to Trigonometry Class 10 ||Maths|| Chapter 8 NCERT Notes

1. Understanding the Concept of Trigonometry

Definition:
Trigonometry comes from the Greek words "trigonon" (meaning triangle) and "metron" (meaning measure). It is the branch of mathematics that deals with the relationships between the angles and sides of a triangle, especially right-angled triangles.
Applications:
Trigonometry is used in various fields like astronomy, navigation, physics, engineering, and architecture, among others. It helps in solving problems where distances or angles are involved.

2. Trigonometric Ratios

In a right-angled triangle, the sides and angles have specific relationships that are defined by the trigonometric ratios. Consider a right-angled triangle ABC where:

Angle $∠ B = 9 0^{\circ}$
$θ$ is one of the acute angles (say $∠ A$ ).

2.1 Trigonometric Ratios of an Acute Angle

In this triangle:

The hypotenuse is the side opposite the right angle.
The adjacent side is the side next to the angle $θ$ .
The opposite side is the side opposite to the angle $θ$ .

The six trigonometric ratios are defined as follows:

Sine (sin θ):
$sin θ = \frac{Opposite side}{Hypotenuse}$
Cosine (cos θ):
$cos θ = \frac{Adjacent side}{Hypotenuse}$
Tangent (tan θ):
$tan θ = \frac{Opposite side}{Adjacent side} = \frac{sin θ}{cos θ}$
Cosecant (csc θ):
$cosec θ = \frac{1}{sin θ} = \frac{Hypotenuse}{Opposite side}$
Secant (sec θ):
$sec θ = \frac{1}{cos θ} = \frac{Hypotenuse}{Adjacent side}$
Cotangent (cot θ):
$cot θ = \frac{1}{tan θ} = \frac{Adjacent side}{Opposite side}$

3. Values of Trigonometric Ratios for Specific Angles

The trigonometric ratios for specific standard angles such as 0°, 30°, 45°, 60°, and 90° are important to memorize. These are often derived using specific triangles or using unit circle concepts.

Angle (θ)	sin θ	cos θ	tan θ	cosec θ	sec θ	cot θ
0°	0	1	0	Undefined	1	Undefined
30°	1/2	√3/2	1/√3	2	2/√3	√3
45°	1/√2	1/√2	1	√2	√2	1
60°	√3/2	1/2	√3	2/√3	2	1/√3
90°	1	0	Undefined	1	Undefined	0

4. Trigonometric Identities

Trigonometric identities are fundamental equations that relate the trigonometric ratios. These hold true for any angle $θ$ . The basic identities you need to know are:

4.1 Fundamental Trigonometric Identity:

Pythagorean Identity: $\sin^{2} θ + \cos^{2} θ = 1$

This identity is derived from the Pythagorean theorem applied to a right-angled triangle.

4.2 Other Important Identities:

From the Pythagorean identity, you can derive the following two important identities:

$1 + {tan}^{2} θ = {sec}^{2} θ$
$1 + {cot}^{2} θ = {cosec}^{2} θ$

5. Complementary Angles

Two angles are said to be complementary if their sum is $9 0^{\circ}$ . The trigonometric ratios of complementary angles are related as follows:

$\sin (9 0^{\circ} - θ) = \cos θ$
$\cos (9 0^{\circ} - θ) = \sin θ$
$\tan (9 0^{\circ} - θ) = \cot θ$
$\cot (9 0^{\circ} - θ) = \tan θ$
$\sec (9 0^{\circ} - θ) = cosec θ$
$cosec (9 0^{\circ} - θ) = \sec θ$

6. Trigonometric Ratios of Complementary Angles

When solving problems involving complementary angles, use the identities mentioned above to simplify the expressions and calculations. These identities help in converting trigonometric ratios of one angle to the other angle in complementary pairs.

7. Use of Trigonometric Tables

In practical applications of trigonometry, especially in earlier times, tables of trigonometric values were used. While today calculators often perform these tasks, knowing the values of trigonometric ratios for specific angles (as shown above) remains essential for exams.

8. Summary of Key Points:

Trigonometry is the study of relationships between the sides and angles of triangles.
Trigonometric ratios relate the angles of a right triangle to the lengths of its sides.
The six trigonometric ratios are: sin, cos, tan, cosec, sec, and cot.
Pythagorean identities and complementary angle identities are crucial for solving trigonometric problems.
Standard angles and their trigonometric values (for 0°, 30°, 45°, 60°, 90°) should be memorized.

That's a detailed breakdown of Chapter 8: Introduction to Trigonometry. This chapter forms the basis for more advanced concepts in trigonometry, which you will explore in higher classes. If you need any more detailed explanations, practice problems, or further help, feel free to ask! ✒️💡

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