CBSE Class 11 Maths Chapter 3: Trigonometric Functions Notes

Understanding Angles and Basic Trigonometric Functions

Angles are a measure of rotation of a given ray about its initial point. The rotation can be positive (anti-clockwise) or negative (clockwise). There are two primary units for measuring angles: degrees and radians.

• Degree Measure: If a rotation from the initial to terminal side is (1/360)th of a revolution, the angle is said to have a measure of one degree (1°). Each degree is divided into 60 minutes (60') and each minute into 60 seconds (60'').
• Radian Measure: An angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle is said to have a measure of one radian (1ᶜ). This unit is particularly useful in calculus and higher mathematics.

Relation between Degree and Radian: The most fundamental conversion is π radians = 180°. From this, 1 radian = 180°/π and 1° = π/180 radians.

The Unit Circle and Trigonometric Ratios

The unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) in the Cartesian coordinate system. It provides a powerful way to define trigonometric functions for any real number (angle). For any point P(a, b) on the unit circle corresponding to an angle θ:

• sin θ = b (y-coordinate)
• cos θ = a (x-coordinate)
• tan θ = b/a (y/x), where a ≠ 0

The reciprocal functions are:

• cosec θ = 1/sin θ = 1/b (b ≠ 0)
• sec θ = 1/cos θ = 1/a (a ≠ 0)
• cot θ = 1/tan θ = a/b (b ≠ 0)

These definitions extend trigonometry beyond acute angles to all real numbers. The sign of these functions depends on the quadrant in which the terminal side of the angle lies. This is often remembered using the CAST rule:

• C (4th Quadrant): Cos and its reciprocal Sec are positive.
• A (1st Quadrant): All trigonometric functions are positive.
• S (2nd Quadrant): Sin and its reciprocal Cosec are positive.
• T (3rd Quadrant): Tan and its reciprocal Cot are positive.

Understanding the unit circle and the CAST rule is vital for accurately determining the values and signs of trigonometric functions for any given angle.

Key Terminology in Trigonometry

Radian Measure

The angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

Unit Circle

A circle with a radius of 1 unit centered at the origin (0,0), used to define trigonometric functions for all real numbers.

Coterminal Angles

Angles that have the same initial and terminal sides. For any angle θ, θ ± 2nπ (or θ ± 360°n) are coterminal angles, where n is an integer.

Periodicity

The property of a function where its values repeat after a fixed interval. For example, sin(x + 2π) = sin x, so the period of sin x is 2π.

Principal Solution

The solution to a trigonometric equation that lies in the interval [0, 2π) or [0°, 360°). There are usually two principal solutions for a given trigonometric value.

General Solution

An expression giving all possible solutions to a trigonometric equation, accounting for the periodic nature of trigonometric functions. It involves an integer 'n'.

Essential Trigonometric Formulas and Identities

Solving Trigonometric Equations: General Solutions

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Worked Examples for Trigonometric Problems

• {"title":"Example 1: Prove the Identity","description":"Prove that (sin x + sin 3x) / (cos x + cos 3x) = tan 2x.\n\nSolution:\nLHS = (sin x + sin 3x) / (cos x + cos 3x)\nUsing sum-to-product formulas:\nsin A + sin B = 2 sin((A+B)/2) cos((A-B)/2)\ncos A + cos B = 2 cos((A+B)/2) cos((A-B)/2)\n\nLHS = (2 sin((x+3x)/2) cos((x-3x)/2)) / (2 cos((x+3x)/2) cos((x-3x)/2))\nLHS = (2 sin(2x) cos(-x)) / (2 cos(2x) cos(-x))\nSince cos(-x) = cos x,\nLHS = (sin 2x) / (cos 2x) = tan 2x = RHS.\nHence Proved."}
• {"title":"Example 2: Find General Solution","description":"Find the general solution of cos x = -1/2.\n\nSolution:\nWe know cos(π/3) = 1/2. Since cos x is negative, x lies in the 2nd or 3rd quadrant.\nFor the 2nd quadrant: x = π - π/3 = 2π/3\nSo, cos x = cos(2π/3)\nUsing the general solution formula for cos x = cos y (which is x = 2nπ ± y):\nx = 2nπ ± 2π/3, where n ∈ Z.\nThis is the general solution."}

Must Remember: Trigonometry Quick Facts

• 1 radian ≈ 57° 16' (approx. 57.3°).
• The range of sin x and cos x is [-1, 1].
• The domain of tan x and sec x is R - {(2n+1)π/2 : n ∈ Z}.
• The domain of cot x and cosec x is R - {nπ : n ∈ Z}.
• sin(nπ) = 0 and cos(nπ) = (-1)ⁿ for any integer 'n'.
• For sin x = k or cos x = k to have a solution, |k| ≤ 1 must hold true.
• Memorize values of sin, cos, tan for common angles: 0, π/6, π/4, π/3, π/2, π.
• Always check for domain restrictions when solving equations; division by zero is a common mistake.

Cracking Trigonometry in Exams

Trigonometry questions often test your ability to apply the correct identity or formula at the right moment. Don't just memorize, understand the derivation for key identities. For proofs, try working from both sides of the equation towards a common expression if one side seems too complex. For trigonometric equations, always look for principal solutions first, then generalize using the appropriate formula. Remember to check for extraneous solutions if you squared both sides of an equation during simplification. Practicing a variety of problems, especially those involving multiple identities, will solidify your understanding and speed.

Quick Check Your Understanding

• Q: Convert 40° 20' into radian measure. A: 40° 20' = 121π/540 radians.
• Q: If sin x = 3/5 and x is in the second quadrant, find cos x and tan x. A: cos x = -4/5, tan x = -3/4.
• Q: What is the general solution for tan x = tan (π/4)? A: x = nπ + π/4, where n ∈ Z.
• Q: Simplify sin(π/2 - x) cos x + cos(π/2 - x) sin x. A: The expression simplifies to cos²x + sin²x = 1.

Frequently Asked Questions