Trigonometry

• Degree Measure: One complete revolution equals $360°$.
• Radian Measure: One radian is the angle subtended at the center of a circle by an arc equal in length to its radius. One complete revolution equals $2\pi$ radians.

Conversion Formulas: \[ \text{Radians} = \frac{\pi}{180°} \times \text{Degrees}, \qquad \text{Degrees} = \frac{180°}{\pi} \times \text{Radians} \]

Arc Length and Sector Area: For an arc of radius $r$ subtending an angle $\theta$ (in radians) at the center: \[ s = r\theta, \qquad A_{\text{sector}} = \frac{1}{2}r^2\theta = \frac{1}{2}rs \]

Trigonometric Functions via Unit Circle: For any angle $\theta$, define a point $P(\cos\theta, \sin\theta)$ on the unit circle ($x^2 + y^2 = 1$) reached by rotating the positive $x$-axis by $\theta$. Then: \[ \sin\theta = y, \quad \cos\theta = x, \quad \tan\theta = \frac{y}{x}\ (x \neq 0) \] \[ \csc\theta = \frac{1}{y}\ (y\neq 0), \quad \sec\theta = \frac{1}{x}\ (x \neq 0), \quad \cot\theta = \frac{x}{y}\ (y \neq 0) \]

Sign in Four Quadrants (ASTC rule): "All Students Take Calculus" — in Q1, All positive; in Q2, Sine positive; in Q3, Tangent positive; in Q4, Cosine positive.

Values at Standard Angles:

Fundamental Pythagorean Identities: \[ \sin^2\theta + \cos^2\theta = 1, \quad 1 + \tan^2\theta = \sec^2\theta, \quad 1 + \cot^2\theta = \csc^2\theta \]

Periodicity: $\sin(\theta + 2\pi) = \sin\theta$, $\cos(\theta + 2\pi) = \cos\theta$, $\tan(\theta + \pi) = \tan\theta$.

• $\sin\theta: \mathbb{R} \to [-1, 1]$
• $\cos\theta: \mathbb{R} \to [-1, 1]$
• $\tan\theta: \mathbb{R} \setminus \{(2n+1)\pi/2\} \to \mathbb{R}$
• $\sec\theta, \csc\theta: \text{domain restricted}, \text{range} = (-\infty, -1] \cup [1, \infty)$

Compound Angle Formulas: \[ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \] \[ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \] \[ \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}, \qquad \cot(A \pm B) = \frac{\cot A \cot B \mp 1}{\cot B \pm \cot A} \]

Double Angle Formulas: \[ \sin 2A = 2\sin A \cos A = \frac{2\tan A}{1 + \tan^2 A} \] \[ \cos 2A = \cos^2 A - \sin^2 A = 1 - 2\sin^2 A = 2\cos^2 A - 1 = \frac{1 - \tan^2 A}{1 + \tan^2 A} \] \[ \tan 2A = \frac{2\tan A}{1 - \tan^2 A} \]

Triple Angle Formulas: \[ \sin 3A = 3\sin A - 4\sin^3 A, \qquad \cos 3A = 4\cos^3 A - 3\cos A, \qquad \tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A} \]

Half-Angle Formulas (in terms of $\cos A$): \[ \sin\frac{A}{2} = \pm\sqrt{\frac{1 - \cos A}{2}}, \qquad \cos\frac{A}{2} = \pm\sqrt{\frac{1 + \cos A}{2}}, \qquad \tan\frac{A}{2} = \pm\sqrt{\frac{1 - \cos A}{1 + \cos A}} \]

Product-to-Sum Formulas: \[ 2\sin A \cos B = \sin(A + B) + \sin(A - B) \] \[ 2\cos A \sin B = \sin(A + B) - \sin(A - B) \] \[ 2\cos A \cos B = \cos(A - B) + \cos(A + B) \] \[ 2\sin A \sin B = \cos(A - B) - \cos(A + B) \]

Sum-to-Product Formulas: \[ \sin C + \sin D = 2\sin\frac{C+D}{2}\cos\frac{C-D}{2}, \qquad \sin C - \sin D = 2\cos\frac{C+D}{2}\sin\frac{C-D}{2} \] \[ \cos C + \cos D = 2\cos\frac{C+D}{2}\cos\frac{C-D}{2}, \qquad \cos C - \cos D = -2\sin\frac{C+D}{2}\sin\frac{C-D}{2} \]

Conditional Identities (when $A + B + C = \pi$): \[ \sin 2A + \sin 2B + \sin 2C = 4\sin A \sin B \sin C \] \[ \cos 2A + \cos 2B + \cos 2C = -1 - 4\cos A \cos B \cos C \] \[ \tan A + \tan B + \tan C = \tan A \tan B \tan C \]

Maximum and Minimum of $a\sin\theta + b\cos\theta$: \[ a\sin\theta + b\cos\theta = \sqrt{a^2 + b^2}\sin(\theta + \alpha), \quad \alpha = \tan^{-1}(b/a) \] Thus, the range is $[-\sqrt{a^2+b^2}, \sqrt{a^2+b^2}]$.

A trigonometric equation is an equation involving trigonometric functions of one or more unknown angles. The solutions in $[0, 2\pi)$ are called principal solutions, while the full family of solutions (containing infinitely many values differing by appropriate periods) is the general solution.

• $\sin\theta = 0 \implies \theta = n\pi$, $n \in \mathbb{Z}$
• $\cos\theta = 0 \implies \theta = (2n+1)\dfrac{\pi}{2}$, $n \in \mathbb{Z}$
• $\tan\theta = 0 \implies \theta = n\pi$, $n \in \mathbb{Z}$
• $\sin\theta = \sin\alpha \implies \theta = n\pi + (-1)^n \alpha$, $n \in \mathbb{Z}$
• $\cos\theta = \cos\alpha \implies \theta = 2n\pi \pm \alpha$, $n \in \mathbb{Z}$
• $\tan\theta = \tan\alpha \implies \theta = n\pi + \alpha$, $n \in \mathbb{Z}$
• $\sin^2\theta = \sin^2\alpha \implies \theta = n\pi \pm \alpha$, $n \in \mathbb{Z}$ (same for $\cos^2$ and $\tan^2$)

• Convert all terms into one trig function (e.g., all $\sin$, or use Pythagorean substitution).
• Use identities such as $\sin 2\theta = 2\sin\theta\cos\theta$ to factor.
• For $a\cos\theta + b\sin\theta = c$, divide by $\sqrt{a^2+b^2}$ and reduce to $\cos(\theta - \alpha) = c/\sqrt{a^2+b^2}$. Solutions exist iff $|c| \leq \sqrt{a^2+b^2}$.
• Watch for extraneous roots introduced by squaring or multiplying — always verify candidate solutions.

Reduction to Quadratic: Equations of the form $a\sin^2\theta + b\sin\theta + c = 0$ are solved by treating $\sin\theta$ as the variable. The same applies for any single trig function via Pythagorean identities.

The standard trigonometric functions are not bijections on their full domains, so inverse functions are defined only on restricted "principal value branches":

Fundamental Identities: \[ \sin^{-1}(-x) = -\sin^{-1}x, \quad \tan^{-1}(-x) = -\tan^{-1}x, \quad \csc^{-1}(-x) = -\csc^{-1}x \] \[ \cos^{-1}(-x) = \pi - \cos^{-1}x, \quad \cot^{-1}(-x) = \pi - \cot^{-1}x, \quad \sec^{-1}(-x) = \pi - \sec^{-1}x \]

Complementary Relations (valid in their domains): \[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \ (x \in [-1, 1]) \] \[ \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \ (x \in \mathbb{R}) \] \[ \sec^{-1}x + \csc^{-1}x = \frac{\pi}{2} \ (|x| \geq 1) \]

Reciprocal Identity: $\tan^{-1}(1/x) = \cot^{-1}x$ for $x > 0$, and $\tan^{-1}(1/x) = \cot^{-1}x - \pi$ for $x < 0$.

Conversion Formulas (for $x \in [0, 1]$): \[ \sin^{-1}x = \cos^{-1}\sqrt{1 - x^2} = \tan^{-1}\frac{x}{\sqrt{1-x^2}} \]

Sum Formulas (constrained): \[ \tan^{-1}x + \tan^{-1}y = \begin{cases} \tan^{-1}\dfrac{x+y}{1-xy}, & \text{if } xy < 1 \\ \pi + \tan^{-1}\dfrac{x+y}{1-xy}, & \text{if } x > 0, y > 0, xy > 1 \\ -\pi + \tan^{-1}\dfrac{x+y}{1-xy}, & \text{if } x < 0, y < 0, xy > 1 \end{cases} \]

\[ \tan^{-1}x - \tan^{-1}y = \tan^{-1}\frac{x - y}{1 + xy} \ \text{(if } xy > -1\text{)} \]

Multiple-Angle Formulas: \[ 2\tan^{-1}x = \begin{cases} \tan^{-1}\dfrac{2x}{1-x^2}, & |x| < 1 \\ \sin^{-1}\dfrac{2x}{1+x^2}, & |x| \leq 1 \\ \cos^{-1}\dfrac{1-x^2}{1+x^2}, & x \geq 0 \end{cases} \]

\[ 3\tan^{-1}x = \tan^{-1}\frac{3x - x^3}{1 - 3x^2} \quad \text{(within appropriate range)} \]

1. Express the equation in a form where you can apply sum/difference formulas.
2. Take sine, cosine, or tangent of both sides to simplify.
3. Always verify candidate solutions because inverse functions have restricted ranges.
4. Watch for the domain restrictions and the conditions under which formulas apply (e.g., $xy < 1$ for $\tan^{-1}x + \tan^{-1}y = \tan^{-1}\dfrac{x+y}{1-xy}$).

Common Simplifications: \[ \sin(\sin^{-1}x) = x \text{ for } x \in [-1, 1] \] \[ \sin^{-1}(\sin x) = x \text{ only if } x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]; \text{ otherwise adjust using periodicity and symmetry} \]

• For expressions with $\sqrt{1 - x^2}$, substitute $x = \sin\theta$ (or $\cos\theta$).
• For expressions with $\sqrt{1 + x^2}$, substitute $x = \tan\theta$.
• For expressions with $\sqrt{x^2 - 1}$, substitute $x = \sec\theta$.

Useful Identities (Standard JEE Tools): \[ \tan^{-1}\frac{x}{1 + xy} = \tan^{-1}x - \tan^{-1}y \cdot \frac{1}{1 + xy} \quad \text{(telescoping)} \]

Telescoping Series: For a sum like $\sum_{k=1}^n \tan^{-1}\dfrac{1}{1 + k + k^2}$, write each term as a telescoping difference: \[ \tan^{-1}\frac{1}{1 + k + k^2} = \tan^{-1}(k+1) - \tan^{-1}k \]

For a triangle $ABC$ with sides $a = BC$, $b = CA$, $c = AB$ opposite to vertices $A$, $B$, $C$:

Sine Rule: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \] where $R$ is the circumradius. The sine rule relates side lengths to opposite angles.

Cosine Rule: \[ \cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{c^2 + a^2 - b^2}{2ca}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]

Projection Formulas: \[ a = b\cos C + c\cos B, \quad b = c\cos A + a\cos C, \quad c = a\cos B + b\cos A \]

Tangent Rule (Napier's Analogy): \[ \tan\frac{B - C}{2} = \frac{b - c}{b + c}\cot\frac{A}{2} \]

Half-Angle Formulas: Let $s = \dfrac{a+b+c}{2}$ be the semi-perimeter. Then: \[ \sin\frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{bc}}, \quad \cos\frac{A}{2} = \sqrt{\frac{s(s-a)}{bc}}, \quad \tan\frac{A}{2} = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}} \]

Area Formulas: \[ \Delta = \frac{1}{2}ab\sin C = \frac{1}{2}bc\sin A = \frac{1}{2}ca\sin B \] \[ \Delta = \sqrt{s(s-a)(s-b)(s-c)} \quad \text{(Heron's formula)} \] \[ \Delta = \frac{abc}{4R} = rs \]

Inradius, Circumradius, Exradii: \[ R = \frac{abc}{4\Delta}, \qquad r = \frac{\Delta}{s} = (s-a)\tan\frac{A}{2} \] \[ r_1 = \frac{\Delta}{s-a} = s\tan\frac{A}{2}, \quad r_2 = \frac{\Delta}{s-b}, \quad r_3 = \frac{\Delta}{s-c} \]

Important Relations: \[ r_1 + r_2 + r_3 - r = 4R \] \[ \frac{1}{r} = \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} \] \[ r_1 r_2 + r_2 r_3 + r_3 r_1 = s^2 \]

Length of Median: The median from vertex $A$ to midpoint of side $a$ has length: \[ m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2} \]

Length of Angle Bisector from $A$: \[ t_a = \frac{2bc\cos(A/2)}{b + c} \]

The applications of trigonometry to heights and distances involve the careful use of angles of elevation and depression.

Angle of Elevation: The angle between the horizontal and the line of sight to an object that is above the observer's eye level.

Angle of Depression: The angle between the horizontal and the line of sight to an object that is below the observer's eye level.

1. Draw a clear diagram identifying all observation points, target points, and key angles.
2. Identify right triangles or general triangles relevant to the problem.
3. Use $\tan\theta$ for ratios of vertical to horizontal distances.
4. For multi-observation problems, set up equations from each observation point and eliminate the unknown.

Bearing Problems (3D): In navigation and aviation, the bearing of point $B$ from point $A$ is the angle measured clockwise from due North (in the horizontal plane) to the line $AB$. In 3D bearing problems, one must combine horizontal angles (bearings) with vertical angles (elevations) using vector geometry or solid trigonometry.

• For an observer at the bottom and a tower of height $h$, with horizontal distance $d$: $\tan(\text{elevation}) = h/d$.
• If two observers are at horizontal distance $d$ apart and observe the top of a tower at angles of elevation $\alpha$ and $\beta$: \[ h = \frac{d \sin\alpha \sin\beta}{\sin(\alpha - \beta)} \quad \text{(observer between tower and other point)} \]