JEE Maths Notes: Chapter on Trigonometric Ratios

1. Introduction to Trigonometric Ratios

Trigonometric ratios are fundamental in understanding the relationship between the angles and sides of a right-angled triangle. These ratios form the basis for trigonometry, which is widely used in solving problems related to angles, distances, and heights in various fields such as physics, engineering, and architecture. The six trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent, and each ratio represents a specific relationship between the sides of a right triangle.

2. Basic Trigonometric Ratios

The six primary trigonometric ratios are defined as follows for a right-angled triangle with an angle θ:

• Sine (sin θ): sin θ = Opposite / Hypotenuse
• Cosine (cos θ): cos θ = Adjacent / Hypotenuse
• Tangent (tan θ): tan θ = Opposite / Adjacent
• Cosecant (csc θ): csc θ = 1 / sin θ = Hypotenuse / Opposite
• Secant (sec θ): sec θ = 1 / cos θ = Hypotenuse / Adjacent
• Cotangent (cot θ): cot θ = 1 / tan θ = Adjacent / Opposite

These ratios provide the foundation for solving trigonometric equations, simplifying expressions, and working with angles in various geometric and real-world contexts.

3. Understanding the Unit Circle

The unit circle is a circle with a radius of 1 unit, centered at the origin of the coordinate plane. It plays a crucial role in defining trigonometric ratios for all angles, not just for those in right triangles. The coordinates of any point on the unit circle can be represented as (cos θ, sin θ), which correspond to the cosine and sine of the angle θ, respectively. The unit circle allows trigonometric functions to be extended to all real numbers and provides a deeper understanding of periodicity and the behavior of these functions.

4. Trigonometric Ratios in Different Quadrants

The values of trigonometric ratios vary depending on the quadrant in which the angle θ lies. The four quadrants of the coordinate plane are divided as follows:

• First Quadrant (0° to 90°): All six trigonometric ratios are positive.
• Second Quadrant (90° to 180°): Sine and cosecant are positive, while cosine, secant, tangent, and cotangent are negative.
• Third Quadrant (180° to 270°): Tangent and cotangent are positive, while sine, cosine, cosecant, and secant are negative.
• Fourth Quadrant (270° to 360°): Cosine and secant are positive, while sine, tangent, cotangent, and cosecant are negative.

Understanding the signs of the trigonometric functions in different quadrants is essential for solving problems involving angles greater than 90° and for determining the correct values in various applications.

5. Trigonometric Ratios of Special Angles

There are specific standard angles for which the values of the trigonometric ratios are well known. These angles include 0°, 30°, 45°, 60°, and 90°. The values for these angles are as follows:

• sin 0° = 0, cos 0° = 1, tan 0° = 0
• sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3
• sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1
• sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
• sin 90° = 1, cos 90° = 0, tan 90° = undefined

These special angle values are fundamental in simplifying trigonometric expressions and solving problems involving angles with exact values.

6. Reciprocal Trigonometric Ratios

The reciprocal trigonometric ratios are obtained by taking the reciprocal of the basic trigonometric functions. These ratios are as follows:

• Cosecant (csc θ): csc θ = 1 / sin θ
• Secant (sec θ): sec θ = 1 / cos θ
• Cotangent (cot θ): cot θ = 1 / tan θ

These ratios are commonly used in problems involving equations and identities, and understanding their relationships with the primary trigonometric ratios is important for simplifying expressions.

7. Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable within a given domain. Some important trigonometric identities include:

• Basic Pythagorean Identity: sin² θ + cos² θ = 1
• Secant and Tangent Identity: sec² θ = 1 + tan² θ
• Cosecant and Cotangent Identity: csc² θ = 1 + cot² θ
• Reciprocal Identities: sin θ = 1/csc θ, cos θ = 1/sec θ, tan θ = 1/cot θ

These identities are crucial for simplifying trigonometric expressions, solving equations, and proving other mathematical relationships.

8. Trigonometric Ratios of Angles Greater than 90°

The trigonometric ratios of angles greater than 90° can be determined by using the properties of the unit circle and the signs of the functions in different quadrants. For example, the sine of 120° is positive, while the cosine of 120° is negative. Understanding these properties is essential for solving problems involving angles beyond the first quadrant.

9. Applications of Trigonometric Ratios

Trigonometric ratios have wide applications in various fields. Some common applications include:

• Physics: Analyzing waveforms, oscillations, and circular motion.
• Engineering: Designing structures, bridges, and buildings, where angles and forces are crucial.
• Navigation: Calculating distances, bearings, and angles in mapping and navigation systems.
• Astronomy: Calculating the positions of celestial bodies and their movement.

10. Practice Problems and Examples

This section includes a variety of practice problems to reinforce the understanding of trigonometric ratios. Problems range from finding the values of trigonometric functions for specific angles, using identities to simplify expressions, to solving real-life problems involving trigonometry. Step-by-step solutions are provided to help students grasp the concepts more effectively.

11. Summary and Key Takeaways

In conclusion, trigonometric ratios are essential tools in mathematics, with widespread applications in geometry, physics, engineering, and more. By understanding the definitions of the six basic trigonometric ratios, their properties, identities, and applications, students can solve a wide variety of problems. Mastery of these concepts is critical for success in JEE and other competitive exams.