Trigonometric Functions revision notes

Introduction to Trigonometry

The word ‘trigonometry’ is derived from the Greek words ‘trigon’ and ‘metron’, which literally means ‘measuring the sides of a triangle’. Originally developed for geometry, navigation, and surveying, it is now used extensively in fields like seismology, electrical circuits, and analyzing musical tones.

Angles and Rotation

An angle is defined as a measure of rotation of a given ray about its initial point (vertex).

Systems of Angle Measurement

Angles are quantified by the amount of rotation from the initial to the terminal side.

Trigonometric Functions via the Unit Circle

Consider a unit circle ($a^2 + b^2 = 1$) centered at the origin. For any point $P(a, b)$ on the circle where the angle substituted from the positive x-axis is $x$ radians:

Historical Note

The study of trigonometry as known today originated in India with mathematicians like Aryabhatta, Brahmagupta, and Bhaskara I and II. The concept of the "sine" of an angle represents the main contribution of Sanskrit astronomical works to mathematics. Thales is historically credited with determining the height of an Egyptian pyramid using shadows (similar triangles).

Arc Length Formula:

• $l = r\theta$, where $l$ is arc length, $r$ is the radius, and $\theta$ is the angle subtended at the center in radians. JEE Tip A massive MCQ trap is plugging $\theta$ in degrees into this formula. Always convert $\theta$ to radians first.

Degree-Radian Conversion: One complete revolution subtends $2\pi$ radians, which equals 360°.

• $\pi \text{ radian} = 180^\circ$.
• $1 \text{ radian} = \frac{180^\circ}{\pi} \approx 57^\circ 16'$.
• $1^\circ = \frac{\pi}{180} \text{ radian} \approx 0.01746 \text{ radian}$.

Domain and Range

Because the coordinates $(a, b)$ on the unit circle are bounded between -1 and 1 ($-1 \le a \le 1$ and $-1 \le b \le 1$), the sine and cosine functions are bounded.

• $y = \sin x$ and $y = \cos x$: Domain = all real numbers ($\mathbb{R}$); Range = $[-1, 1]$.
• $y = \text{cosec } x$: Domain = $\{x \in \mathbb{R} : x \ne n\pi, n \in \mathbb{Z}\}$; Range = $(-\infty, -1] \cup [1, \infty)$.
• $y = \sec x$: Domain = $\{x \in \mathbb{R} : x \ne (2n+1)\frac{\pi}{2}, n \in \mathbb{Z}\}$; Range = $(-\infty, -1] \cup [1, \infty)$. JEE Tip The equations $\sec x = k$ or $\text{cosec } x = k$ have no real solutions if $-1 < k < 1$.
• $y = \tan x$: Domain = $\{x \in \mathbb{R} : x \ne (2n+1)\frac{\pi}{2}, n \in \mathbb{Z}\}$; Range = $\mathbb{R}$.
• $y = \cot x$: Domain = $\{x \in \mathbb{R} : x \ne n\pi, n \in \mathbb{Z}\}$; Range = $\mathbb{R}$.

ASTC Sign Convention

The signs of functions depend on the quadrant in which the terminal side of the angle lies.

• Quadrant I ($0 < x < \frac{\pi}{2}$): All functions are positive.
• Quadrant II ($\frac{\pi}{2} < x < \pi$): Only $\sin x$ and $\text{cosec } x$ are positive.
• Quadrant III ($\pi < x < \frac{3\pi}{2}$): Only $\tan x$ and $\cot x$ are positive.
• Quadrant IV ($\frac{3\pi}{2} < x < 2\pi$): Only $\cos x$ and $\sec x$ are positive.

Periodicity & Quadrantal Angles

Angles that are integral multiples of $\frac{\pi}{2}$ are called quadrantal angles.

• $\sin x = 0$ implies $x = n\pi$ (where $n \in \mathbb{Z}$).
• $\cos x = 0$ implies $x = (2n+1)\frac{\pi}{2}$ (where $n \in \mathbb{Z}$).
• Since one complete revolution brings the point back to its original position, $\sin(2n\pi + x) = \sin x$ and $\cos(2n\pi + x) = \cos x$. JEE Tip To find the value of large angles (e.g., $\sin(1710^\circ)$), continuously subtract $360^\circ$ ($2\pi$) until the angle is within the principal range $[0, 360^\circ]$.

The values of trigonometric functions change continuously as the angle $x$ varies from $0$ to $2\pi$.

• Sine Function: Increases from $0 \to 1$ (Quad I), decreases $1 \to 0$ (Quad II), decreases $0 \to -1$ (Quad III), increases $-1 \to 0$ (Quad IV).
• Cosine Function: Decreases $1 \to 0$ (Quad I), decreases $0 \to -1$ (Quad II), increases $-1 \to 0$ (Quad III), increases $0 \to 1$ (Quad IV).
• Tangent Function: Increases from $0 \to \infty$ in Quad I, increases from $-\infty \to 0$ in Quad II. It repeats after an interval of $\pi$. The statement "$\tan x$ increases from $0 \to \infty$" means it assumes arbitrarily large positive values as $x$ approaches $\frac{\pi}{2}$.
• Secant & Cosecant: Their graphs feature asymptotes where $\cos x = 0$ and $\sin x = 0$, respectively. They never cross the horizontal band between $y = -1$ and $y = 1$.

Basic Fundamental Identities:

• $\cos(-x) = \cos x$ (Even function).
• $\sin(-x) = -\sin x$ (Odd function).
• $\sin^2 x + \cos^2 x = 1$.
• $1 + \tan^2 x = \sec^2 x$.
• $1 + \cot^2 x = \text{cosec}^2 x$.

Sum and Difference of Angles:

• $\cos(x + y) = \cos x \cos y - \sin x \sin y$.
• $\cos(x - y) = \cos x \cos y + \sin x \sin y$.
• $\sin(x + y) = \sin x \cos y + \cos x \sin y$.
• $\sin(x - y) = \sin x \cos y - \cos x \sin y$.
• $\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$.
• $\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$.
• $\cot(x + y) = \frac{\cot x \cot y - 1}{\cot y + \cot x}$.
• $\cot(x - y) = \frac{\cot x \cot y + 1}{\cot y - \cot x}$.
• $\cos(\frac{\pi}{2} - x) = \sin x$ and $\sin(\frac{\pi}{2} - x) = \cos x$.
• $\cos(\pi \pm x) = -\cos x$, $\sin(\pi - x) = \sin x$, $\sin(\pi + x) = -\sin x$.

Multiple Angles (Double and Triple Angles):

• $\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x = \frac{1 - \tan^2 x}{1 + \tan^2 x}$. JEE Tip The rearrangements $1 + \cos 2x = 2\cos^2 x$ and $1 - \cos 2x = 2\sin^2 x$ are arguably the most frequently used substitutions in JEE Calculus (Integration/Limits).
• $\sin 2x = 2\sin x \cos x = \frac{2\tan x}{1 + \tan^2 x}$.
• $\tan 2x = \frac{2\tan x}{1 - \tan^2 x}$.
• $\sin 3x = 3\sin x - 4\sin^3 x$.
• $\cos 3x = 4\cos^3 x - 3\cos x$.
• $\tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}$.

Sum to Product (Factorization) Formulae:

• $\cos x + \cos y = 2\cos\frac{x+y}{2} \cos\frac{x-y}{2}$.
• $\cos x - \cos y = -2\sin\frac{x+y}{2} \sin\frac{x-y}{2}$. JEE Tip Pay attention to the negative sign here! It is a massive source of calculation errors. Alternatively, written as $2\sin\frac{x+y}{2} \sin\frac{y-x}{2}$.
• $\sin x + \sin y = 2\sin\frac{x+y}{2} \cos\frac{x-y}{2}$.
• $\sin x - \sin y = 2\cos\frac{x+y}{2} \sin\frac{x-y}{2}$.

Product to Sum (Defactorization) Formulae:

• $2\cos x \cos y = \cos(x+y) + \cos(x-y)$.
• $-2\sin x \sin y = \cos(x+y) - \cos(x-y)$.
• $2\sin x \cos y = \sin(x+y) + \sin(x-y)$.
• $2\cos x \sin y = \sin(x+y) - \sin(x-y)$.

Standard Derived Values:

• $\sin 15^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}}$.
• $\tan 15^\circ = 2 - \sqrt{3}$.
• $\tan 22.5^\circ = \sqrt{2} - 1$.

Formulas involving tangents, cotangents, secants, and cosecants have strict domain restrictions where denominators become zero:

• $\tan(x + y)$ identity is ONLY valid if none of the angles $x$, $y$, and $(x + y)$ is an odd multiple of $\frac{\pi}{2}$. If $(x+y) = \frac{\pi}{2}$, the right side approaches infinity, meaning you should use $\cot(x+y) = 0$ instead.
• $\tan(x - y)$ identity is ONLY valid if none of the angles $x$, $y$, and $(x - y)$ is an odd multiple of $\frac{\pi}{2}$.
• $\cot(x + y)$ and $\cot(x - y)$ identities are ONLY valid if none of the angles $x$, $y$, $(x+y)$, or $(x-y)$ is a multiple of $\pi$.
• The half-angle substitution $\sin 2x = \frac{2\tan x}{1 + \tan^2 x}$ and $\cos 2x = \frac{1 - \tan^2 x}{1 + \tan^2 x}$ is valid provided $x \ne (2n+1)\frac{\pi}{2}$.

Maximum and Minimum Values

The expression $f(x) = a\sin x + b\cos x + c$ has a maximum value of $c + \sqrt{a^2+b^2}$ and a minimum value of $c - \sqrt{a^2+b^2}$.

• JEE Tip Always ensure the angle '$x$' is exactly the same for both sine and cosine before applying this rule.

Trigonometric Series (Angles in A.P.)

• $\sin \alpha + \sin(\alpha+\beta) + \sin(\alpha+2\beta) + ... + \sin(\alpha+(n-1)\beta) = \frac{\sin(\frac{n\beta}{2})}{\sin(\frac{\beta}{2})} \sin(\alpha + (n-1)\frac{\beta}{2})$
• $\cos \alpha + \cos(\alpha+\beta) + \cos(\alpha+2\beta) + ... + \cos(\alpha+(n-1)\beta) = \frac{\sin(\frac{n\beta}{2})}{\sin(\frac{\beta}{2})} \cos(\alpha + (n-1)\frac{\beta}{2})$

Cosine Product Series

• $\cos A \cos 2A \cos 4A ... \cos(2^{n-1}A) = \frac{\sin(2^n A)}{2^n \sin A}$.

Conditional Identities for Triangles (A + B + C = π)

• $\sin 2A + \sin 2B + \sin 2C = 4\sin A \sin B \sin C$
• $\tan A + \tan B + \tan C = \tan A \tan B \tan C$
• $\cot A \cot B + \cot B \cot C + \cot C \cot A = 1$

• Sign Traps with Square Roots: When solving $\cos^2 x = \frac{16}{25}$, it implies $\cos x = \pm \frac{4}{5}$. Do not blindly assume the positive root. You must check which quadrant $x$ lies in to select the correct sign.
• Half Angle Ambiguity: When finding $\sin\frac{x}{2}$ given $\cos x = -\frac{1}{3}$ and $x$ in Quadrant III, you must narrow the interval. If $\pi < x < \frac{3\pi}{2}$, then dividing by 2 gives $\frac{\pi}{2} < \frac{x}{2} < \frac{3\pi}{4}$. Thus, $\frac{x}{2}$ is in Quadrant II, meaning $\sin\frac{x}{2}$ is positive, but $\cos\frac{x}{2}$ is negative.
• Reference Frame vs Angles: The symbols $\infty$ and $-\infty$ in tangent limits simply specify the type of asymptotic behavior, not actual numerical values. Tangent does not "equal" infinity; it increases arbitrarily.
• $\sin(x+y)$ simplification: Never distribute trigonometric functions! $\sin(x+y) \ne \sin x + \sin y$. It expands into a combination of sines and cosines.

• Series evaluation: Evaluating large sums of trigonometric ratios (like telescoping series using $\tan(A-B)$ or the A.P. series formulas).
• Range of composite trigonometric functions: Utilizing the bounds of $a\sin x + b\cos x$ combined with quadratic formulations (e.g., finding the range of $\sin^4 x + \cos^4 x$).
• Equation counting: Finding the number of solutions of $\sin(\dots) = \cos(\dots)$ in a given interval $[0, 2\pi]$ or $[-2\pi, 2\pi]$ by plotting graphs and counting intersection points.
• Half-angle tangent substitution: Rationalizing integrands in Calculus using $t = \tan(x/2)$.