Trigonometric Functions Class 11 Notes: Formulas & Revision

Key Terms in Trigonometry

Radian

A unit of angle measure, where one radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius. π radians = 180°.

Unit Circle

A circle with a radius of 1 unit, centered at the origin of a Cartesian coordinate system. Its equation is x² + y² = 1.

Trigonometric Functions

Functions of an angle, defined using the coordinates of a point on the unit circle. For a point P(x, y) corresponding to angle θ, cos θ = x and sin θ = y.

Period of a Function

The smallest positive value 'T' for which f(x + T) = f(x) for all x in the domain. For sin x and cos x, the period is 2π; for tan x, it is π.

Principal Solutions

The solutions of a trigonometric equation that lie in the interval [0, 2π). An equation has only a limited number of principal solutions.

General Solution

An expression, usually involving an integer 'n', that gives all possible solutions of a trigonometric equation.

ASTC Rule

A mnemonic to remember the signs of trigonometric functions in the four quadrants: All (I), Sin (II), Tan (III), Cos (IV) are positive.

Understanding Functions via the Unit Circle

In Class 10, trigonometric ratios were limited to acute angles within a right-angled triangle. Class 11 extends this concept to any angle—positive, negative, or greater than 360°—using the unit circle. A unit circle is a circle with a radius of 1 centered at the origin, with the equation x² + y² = 1. Imagine a point P(x, y) on this circle that has rotated by an angle θ from the positive x-axis (counter-clockwise for positive θ, clockwise for negative θ). We define the core trigonometric functions based on the coordinates of this point: cos θ = x and sin θ = y. This simple definition is incredibly powerful. Because P(x, y) is on the circle, we know x² + y² = 1, which immediately gives us the fundamental identity: cos²θ + sin²θ = 1. All other functions can be derived from these: tan θ = y/x = sin θ/cos θ, cot θ = x/y, sec θ = 1/x, and csc θ = 1/y. As the point P moves around the circle, the values of x and y change, defining the values and signs of the trigonometric functions in all four quadrants. This model explains why sin and cos values are periodic and are always between -1 and 1.

Signs of Trigonometric Functions (ASTC Rule)

Must-Remember Formulas

• {"point":"Angle Conversion","description":"π radians = 180°. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π."}
• {"point":"Sum and Difference Formulas","description":"sin(A ± B) = sinA cosB ± cosA sinB. cos(A ± B) = cosA cosB ∓ sinA sinB. tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)."}
• {"point":"Double Angle Formulas","description":"sin(2x) = 2sinxcosx. cos(2x) = cos²x - sin²x = 2cos²x - 1 = 1 - 2sin²x. tan(2x) = (2tanx) / (1 - tan²x)."}
• {"point":"Triple Angle Formulas","description":"sin(3x) = 3sinx - 4sin³x. cos(3x) = 4cos³x - 3cosx."}
• {"point":"Sum-to-Product (C-D Formulas)","description":"sinC + sinD = 2sin((C+D)/2)cos((C-D)/2). sinC - sinD = 2cos((C+D)/2)sin((C-D)/2). cosC + cosD = 2cos((C+D)/2)cos((C-D)/2). cosC - cosD = -2sin((C+D)/2)sin((C-D)/2)."}
• {"point":"Product-to-Sum Formulas","description":"2sinAcosB = sin(A+B) + sin(A-B). 2cosAsinB = sin(A+B) - sin(A-B). 2cosAcosB = cos(A+B) + cos(A-B). 2sinAsinB = cos(A-B) - cos(A+B)."}
• {"point":"General Solutions of Equations","description":"If sin x = sin y ⇒ x = nπ + (-1)ⁿy. If cos x = cos y ⇒ x = 2nπ ± y. If tan x = tan y ⇒ x = nπ + y, where n ∈ Z."}
• {"point":"Domain and Range","description":"sin x, cos x: Domain = R, Range = [-1, 1]. tan x: Domain = R - {(2n+1)π/2}, Range = R. sec x: Domain = R - {(2n+1)π/2}, Range = (-∞, -1] U [1, ∞)."}

Worked Examples

• {"problem":"Find the value of cos(-1710°).","solution":"1. Use cos(-θ) = cos(θ): cos(1710°).\n2. Reduce the angle: 1710° = 4 × 360° + 270°. Since cos has a period of 360°, this simplifies to cos(270°).\n3. cos(270°) = cos(180° + 90°) = -cos(90°) = 0."}
• {"problem":"Prove that (sin(5x) + sin(3x)) / (cos(5x) + cos(3x)) = tan(4x).","solution":"1. Apply Sum-to-Product formulas.\n2. Numerator: sin(5x) + sin(3x) = 2sin((5x+3x)/2)cos((5x-3x)/2) = 2sin(4x)cos(x).\n3. Denominator: cos(5x) + cos(3x) = 2cos((5x+3x)/2)cos((5x-3x)/2) = 2cos(4x)cos(x).\n4. Ratio: (2sin(4x)cos(x)) / (2cos(4x)cos(x)) = sin(4x)/cos(4x) = tan(4x). Hence proved."}
• {"problem":"Find the general solution for 2sin²x + 3cosx = 3.","solution":"1. Convert to a single function (cos x): 2(1-cos²x) + 3cosx = 3.\n2. Form a quadratic equation: 2 - 2cos²x + 3cosx = 3 ⇒ 2cos²x - 3cosx + 1 = 0.\n3. Factorize: (2cosx - 1)(cosx - 1) = 0.\n4. Case 1: cosx = 1. General solution: x = 2nπ.\n5. Case 2: cosx = 1/2. General solution: x = 2nπ ± π/3.\n6. The combined general solutions are x = 2nπ and x = 2nπ ± π/3, where n ∈ Z."}

Board Exam Traps & Tips

General vs. Principal Solutions: Read the question carefully. If it asks for 'solutions', provide the general solution involving 'n'. If it asks for 'principal solutions', only list the values in the interval [0, 2π).

Quadrant Signs: A huge number of marks are lost due to sign errors. When finding cos(A) from sin(A), always check which quadrant A lies in to determine if cos(A) is positive or negative.

Formula Application: Don't just memorize formulas; understand when to use them. For simplification problems, Sum-to-Product and Product-to-Sum formulas are key. For solving equations, double-angle and fundamental identities are often used to convert the equation into a quadratic form of a single trigonometric function.

Practice Questions with Solutions

• What is the value of tan(19π/3)? 19π/3 = 6π + π/3. Since tan has a period of π, tan(6π + π/3) = tan(π/3) = √3.
• If cos x = -1/2 and x lies in the third quadrant, find the value of sin x. In Q3, sin is negative. sin²x = 1 - cos²x = 1 - (1/4) = 3/4. So, sin x = -√3/2.
• What is the general solution of the equation tan(2x) = -cot(x + π/3)? Use -cot(θ) = tan(π/2 + θ). So, tan(2x) = tan(π/2 + x + π/3) = tan(x + 5π/6). The general solution is 2x = nπ + x + 5π/6, which gives x = nπ + 5π/6, where n ∈ Z.
• What is the maximum value of the expression 3cos(x) + 4sin(x)? For an expression of the form acos(x) + bsin(x), the maximum value is √(a² + b²). Here, it is √(3² + 4²) = √25 = 5.

Frequently Asked Questions

Frequently Asked Questions