CBSE Notes Class 10 Maths Chapter 8 Introduction To Trigonometry

One of the most important areas of mathematics is Trigonometry, which deals with the relationships between triangles' angles and sides, particularly those that are right-angled. Trigonometry is introduced in the fundamental ideas of Chapter 8 of the CBSE Class 10 Maths curriculum. In order to provide students with a solid foundation in trigonometry and help them study for tests, it also provides comprehensive explanations of fundamental trigonometric ratios, identities, and applications. It also offers important formulae and solved examples.

1.0CBSE Class 10 Maths Notes Chapter 8 Introduction to Trigonometry - Revision Notes

Important Concepts in Trigonometry

Trigonometric Ratios: These are the ratios of the sides of a right-angled triangle with respect to its angles:

• Sine ($\sin \theta$) : Opposite side/Hypotenuse
• Cosine ($\cos \theta$) : Adjacent side/Hypotenuse
• Tangent ($\tan \theta$) : Opposite side/Adjacent side
• Cosecant ($\csc \theta$) : Hypotenuse/Opposite side
• Secant ($\sec \theta$): Hypotenuse/Adjacent side
• Cotangent ($\cot \theta$) : Adjacent side/Opposite side

Values of Trigonometric Ratios: These ratios have specific values for standard angles 0°, 30°, 45°, 60°, and 90°.

Trigonometric Identities

These are equations involving trigonometric functions that hold true for all angles:

• ${\sin }^2\theta +{\cos }^2\theta =1$
• $1+{\tan }^2\theta ={\sec }^2\theta$
• $1+{\cot }^2\theta ={\csc }^2\theta$

Definitions

Trigonometry: The study of the relationships between the angles and sides of triangles.

Angle of Elevation: The angle formed between the horizontal line and the line of sight when looking upward.

Angle of Depression: The angle formed between the horizontal line and the line of sight when looking downward.

Formulas

Trigonometric Ratios

Reciprocal Relations

• $\sin \theta =\frac{1}{\mathrm{cosec}\theta }\text{ or }\mathrm{cosec}\theta =\frac{1}{\sin \theta }$
• $\cos \theta =\frac{1}{\sec \theta }\text{ or }\sec \theta =\frac{1}{\cos \theta }$
• $\tan \theta =\frac{1}{\cot \theta }\text{ or }\cot \theta =\frac{1}{\tan \theta }$

Trigonometric Identities

• ${\sin }^2\theta +{\cos }^2\theta =1$
• $1+{\tan }^2\theta ={\sec }^2\theta$
• $1+{\cot }^2\theta ={\csc }^2\theta$

2.0Solved Problems

Problem 1: With AB = 25 cm and ACB = 30°, get the lengths of the sides BC and AC in a triangle ABC, right-angled at B, using trigonometric ratios.

Solution:

We must select the ratio between BC and the provided side AB in order to determine the length of the side BC.

Since AB is the side opposite to angle C and BC is the side next to it, as can be seen:

tan C = AB/BC.

tan 30°

25/BC = 1/√3

BC = 25 √3 cm

To find the length of the side AC, we consider

sin 30° = AB/AC

1/2  = 25/AC

AC = 50 cm 

Problem 2: Determine sin A and sec A; Given 15 cot A = 8.

Solution:

Let us assume a right-angled triangle ABC, right-angled at B

Given: 15 cot A = 8

So, cot A = 8/15

We are aware that the cot function is equivalent to the length ratio of the opposing and adjacent sides.

Therefore, $cotA=\frac{Adjacentside}{Oppositeside}$

= AB/BC = 8/15

Let AB be 8k, and BC will be 15k, where k is a positive real number.

According to the Pythagoras theorem, the squares of the hypotenuse side are equal to the sum of the squares of the other two sides of a right triangle, and we get,

AC² = AB² + BC²

Substitute the value of AB and BC

AC² =  (8k)² + (15k)²

AC² = 64k² + 225k²

AC² = 289k²

Therefore, AC = 17k

Now, we have to find the value of sin A and sec A. We know that,

$sin(A)=\frac{Oppositeside}{Hypotenuse}$

Substituting the values of BC and AC and cancelling the constant k in both the numerator and denominator, we obtain

sin A = BC/AC = 15k/17k = 15/17

Therefore, sin A = 15/17

The reciprocal of the cos function, which is the ratio of the hypotenuse side's length to the neighbouring side, is the secant, or sec function.

$Sec(A)=\frac{Hypotenuse}{Adjacentside}$

Substitute the Value of BC and AB and cancel the constant k in both numerator and denominator; we get,

AC/AB = 17k/8k = 17/8

Therefore, sec A = 17/8

Problem 3: Evaluate the value of 2 tan 45° + cos² 30° – sin² 60°.

Solution: Since we know from the trigonometric ratio table given above,

tan 45° = 1

sin 60° = √3/2

cos 30° = √3/2

After putting these values in the given equation:

= 2tan 45° + cos² 30° – sin² 60°

= 2(1) + (√3/2)² – (√3/2)²

= 2 + 0 = 2 (Ans)

Problem 4: If tan 2X = cot (X – 18°), where 2X is an acute angle, find the value of X.

Solution: Given,

tan 2X = cot (X – 18°)

As we know by trigonometric identities,

tan 2X = cot (90° – 2X) (as TanQ = Cot (90-Q))

Substituting the above equation in the given equation, we get;

⇒ cot (90° – 2X) = cot (X – 18°)

From the above equation, we can conclude that “cot” is common on RHS and LHS, hence, both angles are equal.

⇒ 90° – 2X = X – 18°

⇒ 108° = 3X

⇒  X = 108°/3 = 36 (ans)

3.0Tips and Tricks

• Memorize Standard Values: Create a table for trigonometric ratios of 0°, 30°, 45°, 60°, and 90° for quick recall.
• Use Mnemonics: Use phrases like "Some People Have Curly Black Hair" to remember the sequence of sin⁡, cos⁡, tan.
• Practice Visualization: Draw right-angled triangles to visualize the relationships between sides and angles.
• Simplify Problems: Convert complex trigonometric expressions into simpler forms using identities.

4.0Key Features of CBSE Class 10 Maths Notes Chapter 8 Introduction to Trigonometry

• Detailed Explanations: These notes provide step-by-step explanations of basic trigonometric concepts in a manner that is easier for the students to understand the relationship between the angles and sides of triangles.
• Solved Examples: Additionally, there are many solved problems in this collection of notes that demonstrate how trigonometric identities and ratios are used in both mathematical and real-world situations.
• Tables for Quick Revision: Students can easily review and remember the numbers since trigonometric ratios for standard angles are presented in a tabular manner.
• Illustrative Diagrams: Labeled right-angled triangles and graphs are provided to the students as visual aids for an understanding of the geometrical approach to trigonometry.
• Exam-Oriented Content: The content of the notes focuses on high-yield topics and commonest exam questions, so the student will be fully equipped to face whatever trigonometry-related question might appear in their exams.
• Practice Problems: An extensive list of practice problems help in understanding while improving a student's ability to solve problems.
• Tips for Simplification: Important tips and shortcuts have been highlighted to help students understand and perform difficult computations accurately during tests.
• Aligned with CBSE Curriculum: The material adheres to the CBSE syllabus, covering every subject in Chapter 8 in a methodical and thorough manner.
• Real-Life Applications: Examples that use real life situations, including measuring distances and heights, are used to demonstrate how trigonometry can be used in practical contexts.
• Student-Friendly Format: Important concepts and formulas are underlined in the notes, which are easy to read and have been created in a straightforward fashion to facilitate efficient revision. 

These CBSE Class 10 Maths Notes for Chapter 8: Introduction to Trigonometry are a thorough and engaging study guide for students who want to become experts in the important subject. Clear explanations, completed questions, and helpful hints are all included in these notes to ensure comprehensive comprehension and sufficient exam preparation.