Trigonometric Functions revision notes

Key Concepts & Definitions

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).

Initial and Terminal Sides:

The original position of the ray is the initial side, and its final position after rotation is the terminal side.

Sign Convention for Rotation:

If the direction of rotation is anticlockwise, the angle is positive; if it is clockwise, the angle is negative. → [JEE TIP] Always verify the direction of rotation in mechanics and phasor diagrams, as clockwise rotation introduces a negative phase shift.

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

• Degree Measure: If a rotation from the initial to the terminal side is $\frac{1}{360}^{th}$ of a full revolution, the angle has a measure of one degree (1°).
  • 1 degree (1°) = 60 minutes (60').
  • 1 minute (1') = 60 seconds (60").
• Radian Measure: An angle subtended at the center of a unit circle by an arc of length 1 unit is defined as 1 radian. Radians and real numbers are considered one and the same mathematically, mapped directly by wrapping a real number line around the unit circle.

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:

• $\cos x = a$
• $\sin x = b$ From this, the fundamental identity naturally arises: $\sin^2 x + \cos^2 x = 1$.

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).

Measurement & Conversion Formulae

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, Range & Sign of Trigonometric Functions

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 $[0, 360^\circ]$.

Important Graphs & Diagrams

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 $y = 1$.

Conditions & Limitations

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 $x \ne (2n+1)\frac{\pi}{2}$.

JEE Advanced Topics

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

Previous Year JEE Topics

• 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 $t = \tan(x/2)$.

Memory Aids & JEE Traps

• → [JEE TIP] Trap 1 - "The π/4 Identity": Memorize $\tan(\frac{\pi}{4} + x) = \frac{1+\tan x}{1-\tan x}$ and $\tan(\frac{\pi}{4} - x) = \frac{1-\tan x}{1+\tan x}$. This appears repeatedly in coordinate geometry and integration.
• → [JEE TIP] Trap 2 - "The Hidden Difference of Squares": $\sin(x+y)\sin(x-y) = \sin^2 x - \sin^2 y = \cos^2 y - \cos^2 x$. Do NOT write this as $\sin^2(x-y)$.
• → [JEE TIP] Trap 3 - "Degree to Radian oversight": In Calculus, the derivative formula $\frac{d}{dx}(\sin x) = \cos x$ is ONLY true if $x$ is in radians. If $x$ is in degrees, $\frac{d}{dx}(\sin x^\circ) = \frac{\pi}{180}\cos x^\circ$.

Top 10 JEE MCQ Traps

1. Misconception $\rightarrow$ Assuming $\sqrt{\cos^2 x} = \cos x$ universally. Correct Understanding $\rightarrow$ $\sqrt{\cos^2 x} = |\cos x|$. It resolves to $-\cos x$ if $x$ is in the 2nd or 3rd quadrants.
2. Misconception $\rightarrow$ Thinking that since $\sec x$ and $\text{cosec } x$ are reciprocals of sine and cosine, their range is the reciprocal of $[-1, 1]$ (i.e. just bounds). Correct Understanding $\rightarrow$ The range strictly excludes the interval $(-1, 1)$. Values like $\sec x = 0.5$ have zero solutions.
3. Misconception $\rightarrow$ Using $\cos A - \cos B = 2\sin(\frac{A+B}{2})\sin(\frac{A-B}{2})$. Correct Understanding $\rightarrow$ Missing the negative sign! It is $-2\sin(\frac{A+B}{2})\sin(\frac{A-B}{2})$ OR $2\sin(\frac{A+B}{2})\sin(\frac{B-A}{2})$.
4. Misconception $\rightarrow$ Setting $\tan(A+B)$ formula equal to a value without checking the limits. Correct Understanding $\rightarrow$ The formula $\tan(A+B)$ is undefined if $A+B = 90^\circ$. If $A+B = 90^\circ$, directly use $\cot(A+B) = 0$ or $1 - \tan A \tan B = 0$.
5. Misconception $\rightarrow$ The maximum value of $\sin x \cos x$ is 1 because $\sin x$ max is 1 and $\cos x$ max is 1. Correct Understanding $\rightarrow$ Variables are coupled. Multiply and divide by 2 to get $\frac{1}{2}\sin 2x$. The maximum value is $1/2$.
6. Misconception $\rightarrow$ Assuming $a\sin x + b\cos y$ has a max of $\sqrt{a^2+b^2}$. Correct Understanding $\rightarrow$ The variables $x$ and $y$ are independent here. The maximum is simply $|a| + |b|$. The $\sqrt{a^2+b^2}$ rule applies ONLY to $a\sin x + b\cos x$ (same angle).
7. Misconception $\rightarrow$ $\sin^{-1}(\sin(x)) = x$ for all $x$. Correct Understanding $\rightarrow$ True only if $x \in [-\pi/2, \pi/2]$. Otherwise, you must fold $x$ back into the principal domain using reference angles (e.g. $\sin(2\pi - x)$).
8. Misconception $\rightarrow$ In a triangle, if $\sin A = \sin B$, then $A = B$. Correct Understanding $\rightarrow$ It could also be $A = 180^\circ - B$. Always account for the obtuse angle possibility unless other data constraints rule it out.
9. Misconception $\rightarrow$ Solving $\sin x = k$ by only writing the first quadrant principal angle $x = \sin^{-1} k$. Correct Understanding $\rightarrow$ You lose infinite solutions. The correct general solution is $x = n\pi + (-1)^n \sin^{-1} k$.
10. Misconception $\rightarrow$ Canceling $\cos x$ from both sides of the equation $\sin x \cos x = \cos x$ to get $\sin x = 1$. Correct Understanding $\rightarrow$ Never divide by a variable expression without accounting for it being zero. Doing so loses the solutions where $\cos x = 0$ ($x = \pi/2, 3\pi/2$). Factor instead: $\cos x (\sin x - 1) = 0$.