Class 11 Mathematics (041) · Chapter 3 NotesTrigonometric Functions

Angles are measured using two primary units: degrees and radians. A full circle is 360 degrees or 2π radians. Degrees are divided into minutes and seconds, where 1 degree equals 60 minutes and 1 minute equals 60 seconds. Radians are based on the radius of a circle, with 1 radian being the angle subtended by an arc of length equal to the radius. The conversion between degrees and radians is given by 180° = π radians, which allows for easy conversion between the two units.

• 1 degree = 60 minutes, 1 minute = 60 seconds
• 1 radian = arc length / radius
• 180° = π radians

Trigonometric ratios, such as sine, cosine, and tangent, are defined for all real numbers using the unit circle. The unit circle has a radius of 1 and is centered at the origin. For any angle θ, the coordinates of the corresponding point on the unit circle are (cosθ, sinθ). Trigonometric identities, such as sin²θ + cos²θ = 1, are fundamental relationships that hold true for all angles. These identities are essential for simplifying expressions and solving equations involving trigonometric functions.

• sin²θ + cos²θ = 1
• tanθ = sinθ / cosθ
• cotθ = cosθ / sinθ

Trigonometric functions, including sine, cosine, tangent, cotangent, secant, and cosecant, are defined for all real numbers. The domain of these functions is all real numbers, except where the denominator is zero. For example, tanθ is undefined where cosθ = 0. The range of these functions varies: sine and cosine have a range of [-1, 1], while tangent and cotangent have a range of all real numbers. Understanding the domain and range is crucial for graphing and solving trigonometric equations.

• Domain of sinθ and cosθall real numbers
• Range of sinθ and cosθ[-1, 1]
• Domain of tanθall real numbers except where cosθ = 0

The graphs of trigonometric functions are periodic, meaning they repeat at regular intervals. The sine and cosine functions have a period of 2π, while the tangent and cotangent functions have a period of π. The graphs of these functions exhibit various behaviors, such as increasing, decreasing, and oscillating. Understanding the graphs helps in visualizing the behavior of trigonometric functions and solving related problems. For example, the sine function increases from 0 to 1 in the first quadrant and decreases from 1 to 0 in the second quadrant.

• Period of sinθ and cosθ2π
• Period of tanθ and cotθπ
• Graphs exhibit periodic behavior

Trigonometric equations involve trigonometric functions and can be solved using various techniques, such as using identities, graphing, and inverse functions. For example, the equation sinθ = 0.5 has solutions at θ = π/6 + 2πn and θ = 5π/6 + 2πn, where n is any integer. Solving trigonometric equations is essential for finding specific angles and understanding the behavior of trigonometric functions. These solutions are used in various applications, such as physics and engineering.

• Use identities to simplify and solve equations
• Graphing can help visualize solutions
• Inverse functions can find specific angles

Trigonometric functions have numerous real-world applications, such as in physics, engineering, and navigation. For example, they are used to describe the motion of waves, the behavior of electric circuits, and the positions of celestial bodies. Trigonometric functions are also essential in solving problems related to heights and distances, such as finding the height of a building or the distance between two points. Understanding these applications helps in applying trigonometric concepts to real-world situations.

• Describe wave motion and electric circuits
• Find heights and distances
• Used in navigation and astronomy