lecture recording on 06 February 2025 at 19.40.21 PM

Introduction to Trigonometry

• Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles.

• It primarily focuses on right-angled triangles, where one angle is exactly 90 degrees.

Key Concepts

Basic Trigonometric Functions

• Sine (sin): In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.

• Cosine (cos): The cosine is the ratio of the length of the adjacent side to the hypotenuse.

• Tangent (tan): The tangent function is the ratio of the opposite side to the adjacent side.

Reciprocal Functions

• Cosecant (csc): The reciprocal of sine: csc(θ) = 1/sin(θ)

• Secant (sec): The reciprocal of cosine: sec(θ) = 1/cos(θ)

• Cotangent (cot): The reciprocal of tangent: cot(θ) = 1/tan(θ)

The Pythagorean Identity

• The key relationship between the sine and cosine of an angle is given by: sin²(θ) + cos²(θ) = 1

Trigonometric Ratios in Right Angles

Example Ratios

• For a right triangle with an angle θ:

  • If the opposite side measures 3 units and the hypotenuse measures 5 units:

    • sin(θ) = 3/5

    • cos(θ) = adjacent/hypotenuse (for adjacent side length of 4, cos(θ) = 4/5)

    • tan(θ) = sin(θ)/cos(θ) = (3/5)/(4/5) = 3/4

Angles of Elevation and Depression

• Angle of Elevation: The angle formed by the horizontal line and the line of sight to an object above the horizontal.

• Angle of Depression: The angle formed by the horizontal line and the line of sight to an object below the horizontal.

• Important for solving real-world problems involving heights and distances.

Applications of Trigonometry

• Trigonometry is used in various fields such as physics, engineering, astronomy, and architecture.

• It helps in calculating heights of objects, distances between points, and angles of elevation/depression in surveying.

Conclusion

• Understanding the fundamental concepts of trigonometry is crucial for solving problems related to triangles and circles.

• Mastery of trigonometric identities and functions opens up various avenues for practical applications.