Trigonometric Functions
Class 11 Maths • Chapter 03 • Comprehensive Interactive Notes
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: \( \pi \text{ radian} = 180^\circ \)
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.
- \( \sin(x+y) = \sin x \cos y + \cos x \sin y \)
- \( \sin(x-y) = \sin x \cos y - \cos x \sin y \)
- \( \cos(x+y) = \cos x \cos y - \sin x \sin y \)
- \( \cos(x-y) = \cos x \cos y + \sin x \sin y \)
- \( \tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \)
- \( \sin 2x = 2 \sin x \cos x = \frac{2\tan x}{1+\tan^2 x} \)
- \( \cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x \)
- \( \tan 2x = \frac{2\tan x}{1-\tan^2 x} \)
- \( \sin x + \sin y = 2\sin\frac{x+y}{2}\cos\frac{x-y}{2} \)
- \( \sin x - \sin y = 2\cos\frac{x+y}{2}\sin\frac{x-y}{2} \)
- \( \cos x + \cos y = 2\cos\frac{x+y}{2}\cos\frac{x-y}{2} \)
- \( \cos x - \cos y = -2\sin\frac{x+y}{2}\sin\frac{x-y}{2} \)
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: