Class 11 Maths Ch3 Trigonometric Functions Notes

Trigonometry is one of the most ancient branches of mathematics. The word itself is derived from the Greek words trigonon (triangle) and metron (measure).

In earlier classes, you studied trigonometric ratios for acute angles in right-angled triangles. In Class 11 Maths Syllabus , these definitions are extended to arbitrary angles and a much deeper understanding is developed.

These trigonometric notes deal with the definition of trigonometric functions, their properties, values at specific points, graphs, and important identities.

Measurement of Angles

An angle is a measure of rotation of a ray about its initial point. To describe angles, two different systems of measurement are used:

1. Degree Measure (°): The most common system. A complete revolution is divided into 360 equal parts. Thus:
   
   • 1 complete revolution = 360°
   • 1 right angle = 90°
   • 1 straight angle = 180°
   • 1° = 60 minutes (1′)
   • 1′ = 60 seconds (1″)
2. Radian Measure (rad): More fundamental in higher mathematics. An angle in radian measure is defined as follows: If an arc of a circle has length equal to the radius of the circle, then the angle subtended by the arc at the center is said to have a measure of 1 radian.

For a circle of radius r and arc length l:

θ (in radius) = l/r

Since the circumference of a circle is 2πr, one complete revolution corresponds to:

θ = 2πr/r = 2π radians

Hence:

• 1 complete revolution = 360° = 2π radians
• 180° = π radians
• 1° = (π/180) radians
• 1 radian = 180°/π (approximately 57°17’45’’)

Important Note: Radian is the natural unit of angle measurement in mathematics because many formulas (like arc length, area of a sector, etc.) become simpler in radian measure.

Relation between Length of Arc, Radius, and Angle

If θ is the angle subtended at the center of a circle of radius rr by an arc of length ll, then:

θ  = l / r

From this,

• l = r θ
• Area of sector =1/2 r² θ (when θ is in radians)

These simple formulas highlight the importance of radian measure.

Trigonometric Functions of Any Angle

In earlier classes, sine, cosine, tangent, etc., were defined only for acute angles in a right triangle. Now, they are extended for any real angle using the unit circle approach.

Consider a unit circle (circle of radius 1) centered at the origin of the Cartesian plane. Draw an angle θ in standard position (initial side along positive x-axis, vertex at origin, measured anticlockwise for positive angles, clockwise for negative angles). The terminal side of the angle meets the circle at point P(x,y).

By definition:

• cos⁡ θ = x (x-coordinate of P)
• sin⁡ θ = y (y-coordinate of P)
• tan ⁡θ = y/x , x ≠ 0
• cot ⁡θ = x/y,  y ≠ 0
• sec⁡ θ = 1/x , x ≠ 0
• cosec ⁡θ = 1/y , y ≠ 0

Thus, the six trigonometric functions are defined for all real values of θ, except where denominators vanish.

Signs of Trigonometric Functions – ASTC Rule

The unit circle is divided into four quadrants:

• Quadrant I (0 to 90° / 0 to π/2): All trigonometric functions are positive.
• Quadrant II (90° to 180° / π/2 to π): Only sine and cosecant are positive.
• Quadrant III (180° to 270° / π to 3π/2): Only tangent and cotangent are positive.
• Quadrant IV (270° to 360° / 3π/2 to 2π): Only cosine and secant are positive.

This rule is often remembered by ASTC (All, Sine, Tangent, Cosine).

Trigonometric Functions of General Angles

Angles greater than 90° or negative angles can be simplified by expressing them in terms of acute angles.

For example:

• sin⁡(π − θ) = sin ⁡θ
• cos⁡(π − θ) = − cos ⁡θ
• tan⁡(π + θ) = tan⁡ θ

These formulas follow from symmetry of the unit circle.

Trigonometric Identities

Identities are equations true for all values of the variable where they are defined. The three fundamental Pythagorean identities are:

1. sin⁡² θ + cos² θ = 1
2. 1 + tan⁡² θ = sec⁡² θ (valid when θ ≠ π/2 , 3π/2,…)
3. 1 + cot⁡² θ = csc² θ (valid when θ ≠ 0,π,2π,…)

These form the foundation for solving trigonometric equations.

Values of Trigonometric Functions at Specific Angles

Using geometry and the unit circle, we obtain exact values at standard angles:

• 0°, 30°, 45°, 60°, 90° or in radians 0, π/6, π/4, π/3, π/2

For example:

sin 0 = 0, sin π/6 = 1/2, sin π/4 = √2/2, sin π/3 = √3/2, sin π/2 = 1

Similarly, values of cosine, tangent, etc., can be derived. These values must be memorized as they are extensively used in problems.

Similarly, values of cosine, tangent, etc., can be derived. These values must be memorized as they are extensively used in problems.

Domain and Range of Trigonometric Functions

• Sine and Cosine: Domain = all real numbers, Range = [–1, 1]
• Tangent: Domain = ℝ \ {π/2 + nπ, n ∈ ℤ}, Range =  ℝ
• Cotangent: Domain = ℝ \ {nπ, n ∈ ℤ}, Range =  ℝ
• Secant: Domain = ℝ \ {π/2 + nπ, n ∈ ℤ}, Range =  (–∞, –1]∪[1, ∞)
• Cosecant: Domain = ℝ \ {nπ, n ∈ ℤ}, Range =  (–∞, –1]∪[1, ∞)

Periodicity of Trigonometric Functions

Trigonometry functions repeat values after certain intervals:

• Sine and Cosine have period 2π.
• Tangent and Cotangent have period π.
• Secant and Cosecant have period 2π.

Thus:

sin(θ + 2π) = sin θ, cos(θ + 2π) = cos θ, tan(θ + π) = tan θ

Graphs of Trigonometric Functions

Understanding graphs gives a visual meaning of periodicity and range.

• Sine graph: Starts from 0, oscillates between −1 and 1, repeats every 2π.

• Cosine graph: Starts from 1, oscillates between −1 and 1, repeats every 2π.

• Tangent graph: Repeats every π, with vertical asymptotes at θ = π/2 + nπ.

• Cotangent graph: Repeats every π, with asymptotes at θ =n π.

• Secant and Cosecant graphs: Reciprocal of cosine and sine graphs, respectively, with asymptotes where denominator is zero.

Trigonometric Equations

A trigonometric equation is an equation that involves trigonometric functions of a variable.

Examples include:

cos⁡²x + 5 cos ⁡x − 7 = 0 , sin ⁡5x + 3 sin⁡² x = 6

The solutions of such equations depend on the interval in which the variable is considered:

• The solution of a trigonometric equation for variable x within the interval 0 ≤ x ≤ 2π is called the principal solution.
• If the solution involves an integer parameter nn, representing the periodic nature of trigonometric functions, it is called the general solution.

Standard Trigonometric Equations and Their General Solutions

The table below summarizes the general solutions of some common trigonometric equations:

Here, n ∈ ℤ (the set of all integers).

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Derivation

Let us prove the result:

sin x = sin y ⇒ x = nπ + (–1)ⁿy, n element of ℤ

Proof:

1. If sin x = sin y, then sin x – sin y = 0

2. Using the identity sin A – sin B = 2 cos (A+B)/2 sin (A–B)/2, we get:

2 cos (x+y)/2 sin (x-y)/2 = 0

3. This gives two possibilities:

• cos (x+y)/2 = 0, i.e. (x+y)/2 = ((2n+1)π)/2
  ⇒ x + y = (2n + 1)π
  ⇒x = (2n + 1)π – y
• sin (x-y)/2 = 0, i.e. (x-y)/2 = nπ
  ⇒ x – y = 2nπ
  ⇒ x = 2nπ + y

4. Combining both sides:

x = nπ + (–1)ⁿy, n ∈ ℤ

Conclusion

So, my dear friend, if trigonometric equations felt scary at first, I hope these notes made them feel lighter and friendlier. Remember, mathematics is about connecting ideas and practicing calmly until they feel natural. 

Don’t worry if you don’t get everything in one go. Even the smartest mathematicians you know once struggled with the same sin⁡θ and cos⁡θ! What matters is that you keep showing up, keep practicing, and keep reminding yourself, you can do this. 

Whenever you feel overwhelmed, just pause, come back to these trigonometry class 11 notes pdf, and read one section at a time. Trust yourself, and step by step, you’ll turn stress into strength. Remember, you’re not just studying for exams, you’re building skills for life. And yes, you’re doing absolutely amazing already. Keep going, champ! 

FAQs

Q1. What is the difference between trigonometric ratios and trigonometric functions?

Ans. Trigonometric ratios are defined only for acute angles in right-angled triangles, while trigonometric functions extend these ratios to all angles using the unit circle concept.

Q2. What is the principal value of a trigonometric function?

Ans. The principal value is the unique value of an inverse trigonometric function that lies within a fixed range. For example, the principal value of sin⁻¹x lies between −π/2 and π/2.

Q3. What are the domains and ranges of sine, cosine, and tangent functions?

Ans. The domain of sine and cosine is all real numbers and their range is from −1 to 1. The domain of tangent excludes odd multiples of π/2 and its range is all real numbers.

Q4. What are periodic functions? Are trigonometric functions periodic?

Ans. Periodic functions repeat their values after regular intervals. Trigonometric functions are periodic; sine and cosine have a period of 2π, while tangent and cotangent have a period of π.

Q5. What are trigonometric identities?

Ans. Trigonometric identities are equalities involving trigonometric functions that are true for all values in their domains. For example, sin²x + cos²x = 1 is one of the most important identities.