Class 11 Trigonometric Functions Notes & Formulas

1. Angles

Angle is a measure of rotation of a given ray about its initial point. There are two main units of measurement:

Degree Measure: \( 1^{\circ} = 60' \) (minutes), \( 1' = 60'' \) (seconds).
Radian Measure: The angle subtended at the centre by an arc of length 1 unit in a unit circle is said to have a measure of 1 radian.
Relation: \( \pi radian = 180^{\circ} \)

Angle Transformer

Convert Degrees to Radians instantly.

Formula: \( Radian = Degree \times \frac{\pi}{180} \)

2. Signs of Trig Functions

The sign of trigonometric functions depends on the quadrant in which the angle lies.

Rule: "All Silver Tea Cups" (ASTC)

ASTC Quadrant Explorer

Click a quadrant to see positive functions.

Click a quadrant above!

2.1 Trigonometric Functions of Negative Angles

Using symmetry of unit circle:

\( \sin(-x) = -\sin x \) → Odd Function
\( \cos(-x) = \cos x \) → Even Function
\( \tan(-x) = -\tan x \)
\( \cosec(-x) = -\cosec x \)
\( \sec(-x) = \sec x \)
\( \cot(-x) = -\cot x \)

Exam Tip: CBSE often asks to simplify expressions using even-odd nature.

2.2 Trigonometric Ratios of Allied Angles

Angle	Equivalent Ratio
\( \sin(\pi - x) \)	\( \sin x \)
\( \cos(\pi - x) \)	\( -\cos x \)
\( \sin(\frac{\pi}{2} - x) \)	\( \cos x \)
\( \tan(\pi + x) \)	\( \tan x \)

Memory Rule: ASTC decides the sign, reference angle decides the value.

2.3 Domain & Range of Trigonometric Functions

Function	Domain	Range
\( \sin x \)	\( R \)	\( [-1,1] \)
\( \tan x \)	\( x \neq (2n+1)\frac{\pi}{2} \)	\( R \)
\( \sec x \)	\( x \neq (2n+1)\frac{\pi}{2} \)	\( (-\infty,-1] \cup [1,\infty) \)

3. Wave Lab: Graphs

Visualize the periodic nature of Sine, Cosine, and Tangent functions.

4. Identity Vault

Comprehensive list of trigonometric identities.

Sum & Difference of Angles

\( \sin(x+y) = \sin x \cos y + \cos x \sin y \)
\( \sin(x-y) = \sin x \cos y - \cos x \sin y \)
\( \cos(x+y) = \cos x \cos y - \sin x \sin y \)
\( \cos(x-y) = \cos x \cos y + \sin x \sin y \)
\( \tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \)

Double Angle Formulas (2x)

\( \sin 2x = 2 \sin x \cos x = \frac{2\tan x}{1+\tan^2 x} \)
\( \cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x \)
\( \tan 2x = \frac{2\tan x}{1-\tan^2 x} \)

Sum to Product (CD Formulas)

\( \sin x + \sin y = 2\sin\frac{x+y}{2}\cos\frac{x-y}{2} \)
\( \sin x - \sin y = 2\cos\frac{x+y}{2}\sin\frac{x-y}{2} \)
\( \cos x + \cos y = 2\cos\frac{x+y}{2}\cos\frac{x-y}{2} \)
\( \cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2} \)

Half-Angle Formulas (x/2)

\( \sin(\frac{x}{2}) = \pm\sqrt{\frac{1-\cos x}{2}} \)
\( \cos(\frac{x}{2}) = \pm\sqrt{\frac{1+\cos x}{2}} \)
\( \tan(\frac{x}{2}) = \pm\sqrt{\frac{1-\cos x}{1+\cos x}} = \frac{\sin x}{1+\cos x} = \frac{1-\cos x}{\sin x} \)

Properties of Triangles (Sine & Cosine Rules)

Sine Rule: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
Cosine Rule (for \(\cos A\)): \( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \)
Cosine Rule (for \(\cos B\)): \( \cos B = \frac{a^2 + c^2 - b^2}{2ac} \)
Cosine Rule (for \(\cos C\)): \( \cos C = \frac{a^2 + b^2 - c^2}{2ab} \)

5. Trigonometric Equations

For any integer \( n \):

Equation	General Solution
\( \sin x = 0 \)	\( x = n\pi \)
\( \cos x = 0 \)	\( x = (2n+1)\frac{\pi}{2} \)
\( \tan x = 0 \)	\( x = n\pi \)
\( \sin x = \sin y \)	\( x = n\pi + (-1)^n y \)
\( \cos x = \cos y \)	\( x = 2n\pi \pm y \)
\( \tan x = \tan y \)	\( x = n\pi + y \)

5.1 Solutions in a Given Interval

Find solutions of trigonometric equations in \( [0, 2\pi] \).

Example: Solve \( \sin x = \frac{1}{2} \) in \( [0,2\pi] \)

Solutions: \( x = \frac{\pi}{6}, \frac{5\pi}{6} \)

Exam Tip: Always check quadrant and interval limits.

Concept Mastery Quiz

1. \( \sin^2 x + \cos^2 x \) equals:

2. In the 3rd Quadrant, which is positive?

3. The value of \( \sin(180^{\circ}) \) is:

4. \( \cos(2\pi - x) \) is equal to:

5. The range of \( f(x) = \sin x \) is: