Heights and Distances: Trigonometric Ratios and Identities Explained

Heights and Distances

A. Trigonometric Ratios

In right angled \(\Delta ABC\), where \(\angle ABC = \theta\)

• \( \sin\theta = \frac{P}{H} = \frac{AC}{BC} \)
• \( \cos\theta = \frac{B}{H} = \frac{AB}{BC} \)
• \( \tan\theta = \frac{P}{B} = \frac{AC}{AB} \)
• \( \cot\theta = \frac{1}{\tan\theta} = \frac{AB}{AC} \)
• \( \sec\theta = \frac{1}{\cos\theta} = \frac{BC}{AB} \)
• \( \text{cosec}\,\theta = \frac{1}{\sin\theta} = \frac{BC}{AC} \)

B. Trigonometric Identities

• (i) \( \sin^2\theta + \cos^2\theta = 1 \)
• (ii) \( \sec^2\theta - \tan^2\theta = 1 \)
• (iii) \( \text{cosec}^2\theta - \cot^2\theta = 1 \)

Trigonometry Formulas Cheat Sheet for Students

Trigonometry Formulas

A comprehensive and accessible guide to the most important formulas in trigonometry.

Function Relationships

$\sin\theta = \frac{1}{\csc\theta}$

$\csc\theta = \frac{1}{\sin\theta}$

$\cos\theta = \frac{1}{\sec\theta}$

$\sec\theta = \frac{1}{\cos\theta}$

$\tan\theta = \frac{1}{\cot\theta} = \frac{\sin\theta}{\cos\theta}$

$\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}$

Pythagorean Identities

$\sin^2\theta + \cos^2\theta = 1$

$\tan^2\theta + 1 = \sec^2\theta$

$\cot^2\theta + 1 = \csc^2\theta$

Double Angle Formulas

$\sin 2\theta = 2\sin\theta\cos\theta$

$\cos 2\theta = \cos^2\theta - \sin^2\theta$
$= 1 - 2\sin^2\theta$
$= 2\cos^2\theta - 1$

$\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$

Triple Angle Formulas

$\sin 3\theta = 3\sin\theta - 4\sin^3\theta$

$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$

$\tan 3\theta = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$

Opposite Angle Formulas

$\sin(-\theta) = -\sin(\theta)$

$\cos(-\theta) = \cos(\theta)$

$\tan(-\theta) = -\tan(\theta)$

$\cot(-\theta) = -\cot(\theta)$

$\sec(-\theta) = \sec(\theta)$

$\csc(-\theta) = -\csc(\theta)$

Cofunction Formulas (in Quadrant I)

$\sin\theta = \cos(\frac{\pi}{2} - \theta)$

$\cos\theta = \sin(\frac{\pi}{2} - \theta)$

$\tan\theta = \cot(\frac{\pi}{2} - \theta)$

$\cot\theta = \tan(\frac{\pi}{2} - \theta)$

$\sec\theta = \csc(\frac{\pi}{2} - \theta)$

$\csc\theta = \sec(\frac{\pi}{2} - \theta)$

Angle Addition Formulas

$\sin(A+B) = \sin A \cos B + \cos A \sin B$

$\sin(A-B) = \sin A \cos B - \cos A \sin B$

$\cos(A+B) = \cos A \cos B - \sin A \sin B$

$\cos(A-B) = \cos A \cos B + \sin A \sin B$

$\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

$\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

Half Angle Formulas

$\sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{2}}$

$\cos\frac{\theta}{2} = \pm\sqrt{\frac{1+\cos\theta}{2}}$

$\tan\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$
$= \frac{1-\cos\theta}{\sin\theta}$
$= \frac{\sin\theta}{1+\cos\theta}$

Power Reducing Formulas

$\sin^2\theta = \frac{1-\cos 2\theta}{2}$

$\cos^2\theta = \frac{1+\cos 2\theta}{2}$

$\tan^2\theta = \frac{1-\cos 2\theta}{1+\cos 2\theta}$

Product-to-Sum Formulas

$\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$

$\cos A \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]$

$\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$

$\cos A \sin B = \frac{1}{2}[\sin(A+B) - \sin(A-B)]$

Sum-to-Product Formulas

$\sin A + \sin B = 2\sin(\frac{A+B}{2})\cos(\frac{A-B}{2})$

$\sin A - \sin B = 2\sin(\frac{A-B}{2})\cos(\frac{A+B}{2})$

$\cos A + \cos B = 2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2})$

$\cos A - \cos B = -2\sin(\frac{A+B}{2})\sin(\frac{A-B}{2})$

Arc Length

$S = r\theta$

Law of Sines

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Law of Tangents

$\frac{a-b}{a+b} = \frac{\tan[\frac{1}{2}(A-B)]}{\tan[\frac{1}{2}(A+B)]}$

Law of Cosines

$a^2 = b^2 + c^2 - 2bc \cos A$

$b^2 = a^2 + c^2 - 2ac \cos B$

$c^2 = a^2 + b^2 - 2ab \cos C$

TN Samacheer 10 Maths Trigonometry Chapter 6 Hand Written Solutions