Trigonometry Essentials

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Angle in Trigonometry Definition: Rotation of a ray from one position to another along a circle's circumference. Initial Ray: Starting position of the ray. Terminal Ray: Final position of the ray. Positive Angles: Anti-clockwise rotation. Negative Angles: Clockwise rotation. Angle Measurement $1^\circ = 60'$ (minutes) $1' = 60''$ (seconds) Radian Measure Definition: The angle subtended at the center of a circle by an arc equal in length to the radius. Denoted by '$^c$'. $1 \text{ radian} = \left(\frac{180}{\pi}\right)^\circ \approx 57^\circ 16'22''$ $1 \text{ degree} = \left(\frac{\pi}{180}\right)^c \approx 0.01746 \text{ radians}$ Degree to Radian: Multiply by $\frac{\pi}{180}$. E.g., $30^\circ = 30 \times \frac{\pi}{180} = \frac{\pi}{6}$ radians. Radian to Degree: Multiply by $\frac{180}{\pi}$. E.g., $\frac{\pi}{2}$ radians $= \frac{\pi}{2} \times \frac{180}{\pi} = 90^\circ$. Sexagesimal System Expression in the form $x^\circ y' z''$ (x degrees, y minutes, z seconds). Clock Angles Angle between two consecutive digits: $30^\circ$ or $\frac{360^\circ}{12}$. Hour hand subtends: $30^\circ$ in 1 minute. Minute hand subtends: $6^\circ$ in 1 minute. Regular Polygons All interior angles, exterior angles, and sides are equal. Sum of all interior angles. Sum of all exterior angles. Each exterior angle: $\frac{360^\circ}{\text{No. of sides}}$. Each interior angle: $180^\circ - \text{Exterior Angle}$. Relation between Degree and Radian Degree $30^\circ$ $60^\circ$ $90^\circ$ $120^\circ$ $180^\circ$ $360^\circ$ Radian $\frac{\pi}{6}$ $\frac{\pi}{3}$ $\frac{\pi}{2}$ $\frac{2\pi}{3}$ $\pi$ $2\pi$ Arc, Radius and Angle Relation Length of arc $l$ subtending angle $\theta$ (in radians) with radius $r$: $l = r\theta$. r l $\theta$ O Trigonometric Functions Let $(x, y)$ be a point on the terminal ray at distance $r$ from the origin. $\sin x = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}$ $\cos x = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}$ $\tan x = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x}$ $\csc x = \frac{1}{\sin x} = \frac{r}{y}$ $\sec x = \frac{1}{\cos x} = \frac{r}{x}$ $\cot x = \frac{1}{\tan x} = \frac{x}{y}$ Quotient Relations $\tan x = \frac{\sin x}{\cos x}$ $\cot x = \frac{\cos x}{\sin x}$ Pythagorean Relations $\cos^2 x + \sin^2 x = 1$ $\cos^2 x = 1 - \sin^2 x$ $\sin^2 x = 1 - \cos^2 x$ $\sec^2 x - \tan^2 x = 1$ $\sec^2 x = 1 + \tan^2 x$ $\tan^2 x = \sec^2 x - 1$ $\csc^2 x - \cot^2 x = 1$ $\csc^2 x = 1 + \cot^2 x$ $\cot^2 x = \csc^2 x - 1$ Quadrants and Signs of Functions The sign of trigonometric functions depends on the quadrant. X Y X' Y' I Quadrant All ratios are +ve $90 - x$ $360 + x$ II Quadrant sin x, csc x are +ve $90 + x$ $180 - x$ III Quadrant tan x, cot x are +ve $180 + x$ $270 - x$ IV Quadrant cos x, sec x are +ve $270 + x$ $360 - x$ Hint for memory: A ll S tudents T ake C offee (ASTC Rule) I: All ratios positive II: Sine, Cosecant positive III: Tangent, Cotangent positive IV: Cosine, Secant positive Angle Types Acute Angle: $ Obtuse Angle: $> 90^\circ$ Right Angle: $= 90^\circ$ Trigonometric Ratios of Particular Angles Angle $0^\circ$ $30^\circ$ $45^\circ$ $60^\circ$ $90^\circ$ $\sin \theta$ $0$ $\frac{1}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{\sqrt{3}}{2}$ $1$ $\cos \theta$ $1$ $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{1}{2}$ $0$ $\tan \theta$ $0$ $\frac{1}{\sqrt{3}}$ $1$ $\sqrt{3}$ Undefined Compound Angles and Reduction Formulae $\sin(x + y) = \sin x \cos y + \cos x \sin y$ $\sin(x - y) = \sin x \cos y - \cos x \sin y$ $\cos(x + y) = \cos x \cos y - \sin x \sin y$ $\cos(x - y) = \cos x \cos y + \sin x \sin y$ $\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$ $\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$ $\cot(x + y) = \frac{\cot x \cot y - 1}{\cot x + \cot y}$ $\cot(x - y) = \frac{\cot x \cot y + 1}{\cot y - \cot x}$ $\tan(45^\circ + x) = \frac{1 + \tan x}{1 - \tan x}$ $\tan(45^\circ - x) = \frac{1 - \tan x}{1 + \tan x}$ Important Formulae $\sin(x+y)\sin(x-y) = \sin^2 x - \sin^2 y = \cos^2 y - \cos^2 x$ $\cos(x+y)\cos(x-y) = \cos^2 x - \sin^2 y = \cos^2 y - \sin^2 x$ $\tan(x+y)\tan(x-y) = \frac{\tan^2 x - \tan^2 y}{1 - \tan^2 x \tan^2 y}$ Product Formulae (Sum/Difference to Product) $\sin x + \sin y = 2 \sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$ $\sin x - \sin y = 2 \cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)$ $\cos x + \cos y = 2 \cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$ $\cos x - \cos y = -2 \sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)$ Converse of Product Formulae (Product to Sum/Difference) $2 \sin x \cos y = \sin(x+y) + \sin(x-y)$ $2 \cos x \sin y = \sin(x+y) - \sin(x-y)$ $2 \cos x \cos y = \cos(x+y) + \cos(x-y)$ $2 \sin x \sin y = \cos(x-y) - \cos(x+y)$ Reduction Formulae for $(-\theta)$ $\sin(-\theta) = -\sin\theta$ $\cos(-\theta) = \cos\theta$ $\tan(-\theta) = -\tan\theta$ $\csc(-\theta) = -\csc\theta$ $\sec(-\theta) = \sec\theta$ $\cot(-\theta) = -\cot\theta$ Reduction Formulae for $(90^\circ - \theta)$ $\sin(90^\circ - \theta) = \cos\theta$ $\cos(90^\circ - \theta) = \sin\theta$ $\tan(90^\circ - \theta) = \cot\theta$ $\cot(90^\circ - \theta) = \tan\theta$ Reduction Formulae for $(90^\circ + \theta)$ $\sin(90^\circ + \theta) = \cos\theta$ $\cos(90^\circ + \theta) = -\sin\theta$ $\tan(90^\circ + \theta) = -\cot\theta$ $\cot(90^\circ + \theta) = -\tan\theta$ Hint: For $180^\circ \pm \theta$ or $360^\circ \pm \theta$, function remains same. For $90^\circ \pm \theta$ or $270^\circ \pm \theta$, function changes to cofunction. The sign is determined by the quadrant of the original angle. Multiple Angles $\sin 2x = 2 \sin x \cos x = \frac{2 \tan x}{1 + \tan^2 x}$ $\cos 2x = \cos^2 x - \sin^2 x = 1 - 2 \sin^2 x = 2 \cos^2 x - 1 = \frac{1 - \tan^2 x}{1 + \tan^2 x}$ $\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}$ $1 + \cos 2x = 2 \cos^2 x$ $1 - \cos 2x = 2 \sin^2 x$ $1 + \sin 2x = (\cos x + \sin x)^2$ $1 - \sin 2x = (\cos x - \sin x)^2$ $\frac{1 - \cos 2x}{1 + \cos 2x} = \tan^2 x$ $\frac{1 + \sin 2x}{1 - \sin 2x} = \tan^2(45^\circ + x)$ $\sin 3x = 3 \sin x - 4 \sin^3 x$ $\cos 3x = 4 \cos^3 x - 3 \cos x$ $\tan 3x = \frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x}$ Sub-Multiple Angles $\sin x = 2 \sin \frac{x}{2} \cos \frac{x}{2} = \frac{2 \tan \frac{x}{2}}{1 + \tan^2 \frac{x}{2}}$ $\cos x = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2} = 1 - 2 \sin^2 \frac{x}{2} = 2 \cos^2 \frac{x}{2} - 1 = \frac{1 - \tan^2 \frac{x}{2}}{1 + \tan^2 \frac{x}{2}}$ $\tan x = \frac{2 \tan \frac{x}{2}}{1 - \tan^2 \frac{x}{2}}$ $1 + \cos x = 2 \cos^2 \frac{x}{2}$ $1 - \cos x = 2 \sin^2 \frac{x}{2}$ $\frac{1 - \cos x}{1 + \cos x} = \tan^2 \frac{x}{2}$ $\frac{1 + \cos x}{1 - \cos x} = \cot^2 \frac{x}{2}$ $\frac{1 + \sin x}{1 - \sin x} = \tan^2\left(\frac{\pi}{4} + \frac{x}{2}\right)$ $\frac{1 - \sin x}{1 + \sin x} = \tan^2\left(\frac{\pi}{4} - \frac{x}{2}\right)$ Values for $15^\circ$ and $75^\circ$ $\sin 15^\circ = \frac{\sqrt{3}-1}{2\sqrt{2}}$ $\cos 15^\circ = \frac{\sqrt{3}+1}{2\sqrt{2}}$ $\tan 15^\circ = 2 - \sqrt{3}$ $\cot 15^\circ = 2 + \sqrt{3}$ $\sin 75^\circ = \frac{\sqrt{3}+1}{2\sqrt{2}}$ $\cos 75^\circ = \frac{\sqrt{3}-1}{2\sqrt{2}}$ $\tan 75^\circ = 2 + \sqrt{3}$ $\cot 75^\circ = 2 - \sqrt{3}$ Special Angles $\sin 18^\circ = \frac{\sqrt{5}-1}{4}$ $\cos 36^\circ = \frac{\sqrt{5}+1}{4}$ $\cos 18^\circ = \frac{\sqrt{10+2\sqrt{5}}}{4}$ $\sin 36^\circ = \frac{\sqrt{10-2\sqrt{5}}}{4}$ Important Tips for Solving Equations $\sin x = 0 \implies x = n\pi$, for $n \in \mathbb{Z}$ $\tan x = 0 \implies x = n\pi$, for $n \in \mathbb{Z}$ $\cos x = 1 \implies x = 2n\pi$, for $n \in \mathbb{Z}$ $\cos x = 0 \implies x = \frac{(2n+1)\pi}{2}$, for $n \in \mathbb{Z}$ $\cos x = -1 \implies x = (2n+1)\pi$, for $n \in \mathbb{Z}$ Domain and Range of Trigonometric Functions Function $f(x)$ Domain Range $\sin x$ $\mathbb{R}$ $[-1, +1]$ $\cos x$ $\mathbb{R}$ $[-1, +1]$ $\tan x$ $\mathbb{R} - \left\{(2n+1)\frac{\pi}{2} : n \in \mathbb{Z}\right\}$ $\mathbb{R}$ $\csc x$ $\mathbb{R} - \{n\pi : n \in \mathbb{Z}\}$ $(-\infty, -1] \cup [1, \infty)$ $\sec x$ $\mathbb{R} - \left\{(2n+1)\frac{\pi}{2} : n \in \mathbb{Z}\right\}$ $(-\infty, -1] \cup [1, \infty)$ $\cot x$ $\mathbb{R} - \{n\pi : n \in \mathbb{Z}\}$ $\mathbb{R}$ Graphs of Trigonometric Functions $y = \sin x$ 1 -1 $-\pi$ $\pi$ 0 $y = \cos x$ 1 -1 $-\pi$ $\pi$ 0 $y = \tan x$ $-\pi/2$ $\pi/2$ 0 $y = \cot x$ $\pi$ $\pi/2$ $3\pi/2$ 0