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Class 11 Maths Notes

Trigonometric Functions Class 11 Notes

Complete, exam-ready notes on trigonometric functions: angle systems and degree–radian conversion, arc length, the six trigonometric ratios and their quadrant signs, identities, compound and double-angle formulas, trigonometric equations with general solutions, and the sine and cosine rules — written for CBSE, JEE and NEET revision.

Class11SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is this chapter about in one line?

Trigonometric functions relate angles to ratios of sides, and this chapter builds the identities, compound-angle and double-angle formulas needed to solve equations and triangles.

Angles and Degree–Radian Conversion

Degree and radian

One radian is the angle subtended at the centre by an arc equal in length to the radius. A full revolution is 360° = 2π radians, so 180° = π radians. To convert, multiply degrees by π/180 or multiply radians by 180/π.

1∘=π180 rad,1 rad=180∘π1^\circ = \frac{\pi}{180}\,\text{rad},\qquad 1\,\text{rad} = \frac{180^\circ}{\pi}
Degree–radian conversion

Arc length

l=rθl = r\theta

The length l of an arc of a circle of radius r is l = rθ, where θ is the subtended angle measured in radians (not degrees).

The Six Ratios and Quadrant Signs

Six trigonometric ratios

For an angle θ, the six ratios are sin, cos, tan, cosec (1/sin), sec (1/cos) and cot (1/tan). Their signs depend on the quadrant of the terminal side.

  • Quadrant I: all six ratios are positive.
  • Quadrant II: only sin and cosec positive.
  • Quadrant III: only tan and cot positive.
  • Quadrant IV: only cos and sec positive.
  • Mnemonic: ASTC — All, Sin, Tan, Cos (anti-clockwise from Quadrant I).

Basic Trigonometric Identities

Pythagorean identities

These follow from the unit circle and are true for every angle θ. Dividing sin² + cos² = 1 by cos² or sin² gives the other two.

sin⁡2θ+cos⁡2θ=1,1+tan⁡2θ=sec⁡2θ,1+cot⁡2θ=csc⁡2θ\sin^2\theta + \cos^2\theta = 1,\qquad 1 + \tan^2\theta = \sec^2\theta,\qquad 1 + \cot^2\theta = \csc^2\theta
Pythagorean identities

Compound, Double and Half-Angle Formulas

Compound angles

Formulas for sin, cos and tan of A ± B expand a single angle into two. Setting A = B gives the double-angle identities, from which half-angle and power-reduction formulas follow.

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B
Sine of a sum/difference
cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B
Cosine of a sum/difference
tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}
Tangent of a sum/difference

Sum-to-Product and Product-to-Sum

Product-to-sum

Products of sines and cosines convert into sums of sines or cosines. These are used to integrate and to simplify expressions.

2sin⁡Acos⁡B=sin⁡(A+B)+sin⁡(A−B)2\sin A \cos B = \sin(A+B) + \sin(A-B)
Product-to-sum
sin⁡2A=2sin⁡Acos⁡A\sin 2A = 2\sin A \cos A
Double angle (sine)

Trigonometric Equations and General Solutions

General solution

Sine, cosine and tangent repeat every period, so an equation like sinθ = sinα has infinitely many solutions captured by a periodic formula.

sin⁡θ=sin⁡α  ⟹  θ=nπ+(−1)nα, n∈Z\sin \theta = \sin \alpha \implies \theta = n\pi + (-1)^n \alpha,\ n \in \mathbb{Z}
General solution for sine

Cosine and tangent general solutions

For cosθ = cosα, θ = 2nπ ± α. For tanθ = tanα, θ = nπ + α, for n an integer.

Solved Examples

Example: Convert 120° to radians and 2.5 radians to degrees.

Solution: 120° × π/180 = 2π/3 radians. 2.5 × 180/π ≈ 143.2°.

Example: Solve sinθ = 1/2 for the general solution.

Solution: sinθ = sin(π/6), so θ = nπ + (−1)ⁿ·(π/6) for n ∈ ℤ.

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Arc length

l=rθl = r\theta

Sine of sum/difference

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B

Cosine of sum/difference

cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B

Tangent of sum/difference

tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}

Product-to-sum

2sin⁡Acos⁡B=sin⁡(A+B)+sin⁡(A−B)2\sin A \cos B = \sin(A+B) + \sin(A-B)

Sine general solution

sin⁡θ=sin⁡α  ⟹  θ=nπ+(−1)nα\sin\theta = \sin\alpha \implies \theta = n\pi + (-1)^n \alpha

Sine rule

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

  • 180° = π radians; multiply by π/180 to convert degrees to radians.
  • Arc length l = rθ needs θ in radians.
  • ASTC gives quadrant signs: All, Sin, Tan, Cos.
  • sin²θ + cos²θ = 1 is the base of the other two identities.
  • Double angle: sin2A = 2sinAcosA.
  • General solution of sinθ = sinα is θ = nπ + (−1)ⁿα.
  • Sine rule links sides and angles of any triangle.

FAQ

Common questions

How do you convert degrees to radians?

Multiply the angle in degrees by π/180. Since 180° = π radians, 120° becomes 2π/3 radians and a full circle is 2π radians.

What are the compound-angle formulas?

They expand sin(A±B), cos(A±B) and tan(A±B) into single-angle terms, for example sin(A+B) = sinAcosB + cosAsinB. Setting A = B gives the double-angle formulas.

What is the general solution of sinθ = sinα?

θ = nπ + (−1)ⁿα for any integer n. Since sine repeats every 2π and is symmetrical about π/2, this single formula captures all solutions.

What are the sine and cosine rules?

The sine rule a/sinA = b/sinB = c/sinC relates sides to opposite angles. The cosine rule, c² = a² + b² − 2ab·cosC, finds a side or angle in any triangle.

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