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Class 11 — Mathematics (NCERT)

Trigonometric Functions

Class 11

  • ✓By the end of this lesson students will be able to understand and convert between degree measure and radian measure of angles.
  • ✓By the end of this lesson students will be able to apply fundamental trigonometric identities and compound angle formulae to simplify expressions and prove other identities.
  • ✓By the end of this lesson students will be able to derive and use transformation formulae (product-to-sum and sum-to-product) and multiple/submultiple angle formulae.
  • ✓By the end of this lesson students will be able to find the general solutions of trigonometric equations.

Key concepts

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 called a radian. It is a unit of angle measurement in the S.I. system. The relation between degree and radian measure is fundamental for understanding angles in higher mathematics.

π radians = 180°\nTo convert degrees to radians: Multiply by π/180\nTo convert radians to degrees: Multiply by 180/π\nLength of arc (l) = rθ, where θ is in radians
Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables for which the functions are defined. These identities are crucial for simplifying expressions, solving equations, and proving other mathematical statements.

Fundamental Identities:\nsin²x + cos²x = 1\n1 + tan²x = sec²x\n1 + cot²x = cosec²x\n\nCompound Angle Formulae:\ncos(A + B) = cos A cos B - sin A sin B\ncos(A - B) = cos A cos B + sin A sin B\nsin(A + B) = sin A cos B + cos A sin B\nsin(A - B) = sin A cos B - cos A sin B\ntan(A + B) = (tan A + tan B) / (1 - tan A tan B)\ntan(A - B) = (tan A - tan B) / (1 + tan A tan B)\n\nTransformation Formulae (Product-to-Sum):\n2 cos A cos B = cos(A + B) + cos(A - B)\n2 sin A sin B = cos(A - B) - cos(A + B)\n2 sin A cos B = sin(A + B) + sin(A - B)\n2 cos A sin B = sin(A + B) - sin(A - B)\n\nTransformation Formulae (Sum-to-Product):\nsin C + sin D = 2 sin((C+D)/2) cos((C-D)/2)\nsin C - sin D = 2 cos((C+D)/2) sin((C-D)/2)\ncos C + cos D = 2 cos((C+D)/2) cos((C-D)/2)\ncos C - cos D = -2 sin((C+D)/2) sin((C-D)/2)\n\nMultiple Angle Formulae:\nsin 2x = 2 sin x cos x = (2 tan x) / (1 + tan²x)\ncos 2x = cos²x - sin²x = 2 cos²x - 1 = 1 - 2 sin²x = (1 - tan²x) / (1 + tan²x)\ntan 2x = (2 tan x) / (1 - tan²x)\nsin 3x = 3 sin x - 4 sin³x\ncos 3x = 4 cos³x - 3 cos x\ntan 3x = (3 tan x - tan³x) / (1 - 3 tan²x)
General Solutions of Trigonometric Equations

The solution set of a trigonometric equation containing all possible values of the variable is called the general solution. These solutions are expressed using an integer 'n' to represent the periodicity of trigonometric functions.

If sin x = 0, then x = nπ, where n ∈ Z.\nIf cos x = 0, then x = (2n + 1)π/2, where n ∈ Z.\nIf tan x = 0, then x = nπ, where n ∈ Z.\nIf sin x = sin y, then x = nπ + (-1)ⁿy, where n ∈ Z.\nIf cos x = cos y, then x = 2nπ ± y, where n ∈ Z.\nIf tan x = tan y, then x = nπ + y, where n ∈ Z.\nIf sin²x = sin²y, then x = nπ ± y, where n ∈ Z.\nIf cos²x = cos²y, then x = nπ ± y, where n ∈ Z.\nIf tan²x = tan²y, then x = nπ ± y, where n ∈ Z.

Key facts to remember

  • 1π radians = 180°.
  • 21 radian is approximately 57°17'45'' (or 57.2958°).
  • 3The three fundamental trigonometric identities are sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.
  • 4The general solution for sin x = sin y is x = nπ + (-1)ⁿy, where n ∈ Z.
  • 5The general solution for cos x = cos y is x = 2nπ ± y, where n ∈ Z.
  • 6The general solution for tan x = tan y is x = nπ + y, where n ∈ Z.
  • 7The period of sin x and cos x is 2π, while the period of tan x is π.
  • 8The length of an arc (l) of a circle with radius (r) subtending an angle θ at the centre is given by l = rθ, where θ must be in radians.

Worked examples

Example 1

Convert 40° 20' into radian measure.

IFirst, convert the minutes part into degrees: 20' = (20/60)° = (1/3)°.
IISo, 40° 20' can be written as (40 + 1/3)° = (121/3)°.
IIITo convert degrees to radians, we multiply by π/180.
IVRadian measure = (121/3) × (π/180) radians.
VRadian measure = 121π/540 radians.

Answer

121π/540 radians

Remember that 1° = 60' (minutes) and 1' = 60'' (seconds).

Example 2

Prove that (sin 5x + sin 3x) / (cos 5x + cos 3x) = tan 4x.

IConsider the L.H.S.: (sin 5x + sin 3x) / (cos 5x + cos 3x).
IIApply the sum-to-product formulae:
IIIFor the numerator: sin C + sin D = 2 sin((C+D)/2) cos((C-D)/2)
IVsin 5x + sin 3x = 2 sin((5x+3x)/2) cos((5x-3x)/2) = 2 sin(8x/2) cos(2x/2) = 2 sin 4x cos x.
VFor the denominator: cos C + cos D = 2 cos((C+D)/2) cos((C-D)/2)
VIcos 5x + cos 3x = 2 cos((5x+3x)/2) cos((5x-3x)/2) = 2 cos(8x/2) cos(2x/2) = 2 cos 4x cos x.
VIISubstitute these back into the L.H.S.: (2 sin 4x cos x) / (2 cos 4x cos x).
VIIICancel out the common terms '2' and 'cos x' (assuming cos x ≠ 0).
9L.H.S. = sin 4x / cos 4x = tan 4x.
10This is equal to the R.H.S.

Answer

L.H.S. = R.H.S., hence proved.

Example 3

Find the general solution of the equation sin 2x - sin 4x + sin 6x = 0.

IRearrange the terms to group for sum-to-product formula: (sin 6x + sin 2x) - sin 4x = 0.
IIApply the sum-to-product formula sin C + sin D = 2 sin((C+D)/2) cos((C-D)/2) to (sin 6x + sin 2x):
IIIsin 6x + sin 2x = 2 sin((6x+2x)/2) cos((6x-2x)/2) = 2 sin(8x/2) cos(4x/2) = 2 sin 4x cos 2x.
IVSubstitute this back into the equation: 2 sin 4x cos 2x - sin 4x = 0.
VFactor out sin 4x: sin 4x (2 cos 2x - 1) = 0.
VIThis implies either sin 4x = 0 or 2 cos 2x - 1 = 0.
VIICase 1: sin 4x = 0.
VIIIUsing the general solution for sin θ = 0, we have 4x = nπ, where n ∈ Z.
9Therefore, x = nπ/4, where n ∈ Z.
10Case 2: 2 cos 2x - 1 = 0.
11This gives cos 2x = 1/2.
12We know that cos(π/3) = 1/2. So, cos 2x = cos(π/3).
13Using the general solution for cos θ = cos α, we have θ = 2mπ ± α, where m ∈ Z.
14So, 2x = 2mπ ± π/3, where m ∈ Z.
15Dividing by 2, we get x = mπ ± π/6, where m ∈ Z.

Answer

The general solutions are x = nπ/4 and x = mπ ± π/6, where n, m ∈ Z.

Always factorise trigonometric equations to avoid losing solutions by dividing by a variable term.

Common mistakes

  • ✗Confusing degree and radian measures, especially when using calculators; always ensure the calculator is in the correct mode.
  • ✗Incorrectly applying the signs in compound angle formulae (e.g., writing cos(A+B) = cos A cos B + sin A sin B instead of cos A cos B - sin A sin B).
  • ✗Forgetting to include the integer 'n' or the '(-1)ⁿ' factor in general solutions, leading to incomplete solution sets.
  • ✗Dividing by a trigonometric function (e.g., sin x) without considering the case where that function might be zero, which can lead to loss of valid solutions. Always factorise instead.
  • ✗Not checking for domain restrictions when solving equations involving functions like tan x, sec x, cosec x, cot x, where the denominator cannot be zero.

Exam tips

  • ★Memorise all fundamental trigonometric identities, compound angle formulae, transformation formulae, and multiple/submultiple angle formulae thoroughly. Write them down repeatedly.
  • ★Practice converting angles between degree and radian measures frequently to avoid errors and build confidence.
  • ★When proving identities, always start from one side (L.H.S. or R.H.S.) and systematically transform it to the other side, showing all steps clearly and logically.
  • ★For general solutions, factorise trigonometric expressions whenever possible to break down complex equations into simpler forms. Avoid dividing by variable trigonometric terms.
  • ★Always write 'n ∈ Z' (where n is an integer) when stating general solutions to indicate that 'n' can be any integer.

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