1. 📐 Introduction to Trigonometry

Trigonometry is the study of relationships between angles and sides, and in Class 11 it is extended from simple right-triangle ratios to functions of a real variable. This makes it possible to work with positive, negative, acute, obtuse and angles greater than 360°.

The central idea is to associate every angle with definite numerical values of trigonometric functions. These ideas are useful in geometry, navigation, engineering, physics, astronomy, waves, periodic motion and many other applications.

2. 🔄 Angles and Their Measurement

An angle is formed when a ray rotates about a fixed point. The fixed point is the vertex. The ray before rotation is the initial side, and the ray after rotation is the terminal side.

Positive and Negative Angles

• Positive angle: measured by counterclockwise rotation.
• Negative angle: measured by clockwise rotation.
• Angles can be greater than one complete revolution.
• Angles differing by a complete revolution have the same terminal side and are called coterminal angles.

Quadrants

In standard position, the initial side lies along the positive x-axis. The terminal side determines the quadrant. This becomes important because the signs of sin, cos, tan and their reciprocals depend on the quadrant.

3. 🟠 Radian Measure

A radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.

Here l is the arc length, r is the radius and θ is measured in radians. Therefore:

Degree–Radian Relationship

Conversions

Examples

• 60° = 60π/180 = π/3
• 45° = π/4
• 30° = π/6
• π/2 = 90°
• 2π/3 = 120°

4. 🎯 Trigonometric Functions

For a right-angled triangle and an acute angle θ, the three fundamental ratios are sine, cosine and tangent. In Class 11, these ideas are extended to general angles through the unit circle.

The Six Trigonometric Functions

• sin θ = perpendicular / hypotenuse
• cos θ = base / hypotenuse
• tan θ = perpendicular / base = sin θ / cos θ
• cosec θ = 1 / sin θ
• sec θ = 1 / cos θ
• cot θ = 1 / tan θ = cos θ / sin θ

Domains and Ranges

• sin x: Domain = all real numbers; Range = [−1, 1]
• cos x: Domain = all real numbers; Range = [−1, 1]
• tan x: Domain = all real numbers except π/2 + nπ; Range = all real numbers
• cot x: Domain = all real numbers except nπ; Range = all real numbers
• sec x: Domain = all real numbers except π/2 + nπ; Range = (−∞,−1] ∪ [1,∞)
• cosec x: Domain = all real numbers except nπ; Range = (−∞,−1] ∪ [1,∞)

Here n ∈ Z.

5. ⭕ Unit Circle Definition

The unit circle is a circle of radius 1 centred at the origin. If a point P on the unit circle corresponds to angle θ and has coordinates (x, y), then:

Therefore, directly from the coordinates: cos θ = x and sin θ = y. This definition works for every real angle.

6. ⭐ Standard Angles and Exact Values

You should know the exact values of the trigonometric functions for the standard angles 0°, 30°, 45°, 60° and 90°, and use symmetry/signs to obtain values in other quadrants.

7. 🗺️ Signs of Trigonometric Functions in Quadrants

The sign of a trigonometric function depends on the quadrant containing the terminal side of the angle. The easiest memory aid is ASTC.

• Quadrant I: all six functions are positive.
• Quadrant II: sin and cosec are positive.
• Quadrant III: tan and cot are positive.
• Quadrant IV: cos and sec are positive.

Reference Angle Idea

For a general angle, first find its related acute/reference angle and then apply the correct sign according to the quadrant. This is especially useful for exact values such as sin 150°, cos 210°, tan 315°, etc.

8. 📈 Graphs of Trigonometric Functions

Trigonometric functions are periodic. Their graphs repeat after a fixed interval called the period.

Sine Graph: y = sin x

• Domain: all real numbers
• Range: [−1, 1]
• Period: 2π
• Maximum value = 1; minimum value = −1

Cosine Graph: y = cos x

• Domain: all real numbers
• Range: [−1, 1]
• Period: 2π
• Maximum value = 1; minimum value = −1

Tangent Graph: y = tan x

• Domain: all real numbers except π/2 + nπ
• Range: all real numbers
• Period: π
• Vertical asymptotes occur at x = π/2 + nπ

9. 🛡️ Fundamental Trigonometric Identities

An identity is an equation that is true for every value of the variable for which both sides are defined. These identities are among the most important tools in Class 11 trigonometry.

Pythagorean Identity

Derived Identities

Reciprocal Relations

Quotient Relations

10. ➕➖ Trigonometric Functions of Sum and Difference of Two Angles

These formulas let us find trigonometric values of angles such as 75°, 15° or 105° by expressing them as sums or differences of familiar angles.

Sine of Sum and Difference

Cosine of Sum and Difference

Tangent of Sum and Difference

Example: Find sin 75°

Write 75° = 45° + 30°. Then apply the sine-sum formula:

Example: Find cos 15°

Write 15° = 45° − 30°. Then:

11. 🔥 Important Consequences

The sum-and-difference formulas generate several high-value results used throughout Class 11 and later mathematics.

Double-Angle Formulas

Useful Alternate Forms

12. 🧠 One-Page Exam Revision

Must Know Definitions

• Initial side, terminal side and vertex of an angle.
• Positive and negative angles.
• Coterminal angles.
• Radian measure and arc-length relation.
• Unit circle and general-angle definition of trigonometric functions.
• Domain and range of all six trigonometric functions.
• Periodicity and basic graphs.
• Trigonometric identities.
• Sum and difference formulas.

Must Memorise

Common Mistakes to Avoid

• Forgetting whether the angle is in degrees or radians.
• Using the wrong sign in the quadrant.
• Writing tan θ when cos θ = 0.
• Confusing reciprocal functions with quotient functions.
• Mixing the signs in sin(A ± B) and cos(A ± B).
• Forgetting that tangent has period π, while sine and cosine have period 2π.
• Dividing an identity by a quantity that can be zero without checking its domain.

13. 🏗️ Height & Distance — Existing Interactive Lab

This original interactive section is retained so the rest of the Math Verse experience remains unchanged. It demonstrates how an angle of elevation can be related to a known distance and height.