Trigonometry (Continued)

Introduction

     An equation containing the trigonometrical functions of an unknown quantity is termed as a trigonometrical equation, which holds for some values (and not for all values) of the quantities involved.

Co-terminal Angles

     The angles (2p + A), (4p + A)......have the same initial and terminal arm as the angle A and so these angles are called co-terminal angles. For all such angles, the value of any trigonometric ratio is the same. Thus, all co-terminal angles are trigonometrically equivalent.

Theorem 1

     General solution of sin q = k.

Theorem 2

     General solution of cos q = k.

Theorem 3

     General solution of tan q = k.

Theorem 4

     General solution of acosq + bsinq = c.

Sine Formula (Sine Rule)

     Prove that in any triangle, the sides of a triangle are proportional to the sines of the opposite angles.

Cosine Formula (Cosine Rule)

     If ABC is a triangle with sides a = BC, b = CA, c = AB, then prove that

     a² = b² + c² - 2 bc cos A, b² = c² + a² - 2 ca cos B, c² = a² + b² - 2 ab cos C.

Projection Formula

     In any triangle ABC, with sides a = BC, b = CA and c = BA, then prove that
a = b cos C + c cos B, b = c cos A +    a cos C,  c = a cos B + b cos A

Napier's Analogy

     In any triangle ABC, with sides a = BC, b = CA and c = BA, then prove that

     

     

     

Area of a triangle

     In any triangle ABC, prove that the area of a triangle ABC is given by

     

Hero's Formula

     

Half-Angle Formulae

     If ABC is a triangle with sides a = BC, b = CA, c = AB, then prove that

     

     

     where 2s = a + b + c.

Solution of triangles

     From geometry, we know that when any three elements are given of which necessarily a side is given, the triangle is completely determined i.e, remaining three elements can be determined. The process of determining the unknown elements knowing the known elements is known as the solution of a triangle.

Heights and Distances

     One of the main use of trigonometry is to find the distances between points, or the heights of objects, without actually measuring these distances or these heights.

Inverse Trigonometric Functions

     For x Î [-1, 1], if q is an angle whose sine is x, then we say that sine inverse x is q and write sin^-1x = q.

Hyperbolic Function

We define (e^x+e^{- x})/2 as sinhx and (e^x-e^-x)/2 as coshx. Based on this other trigonometric functions are defined and they are all called hyperbolic functions.

Inverse Hyperbolic functions

The inverse function arsinhx, artanhx, arcschx and arcothx exist for all x є (-∞, ∞) while arcoshx and arsechx exist only for x є (0, ∞).

Summary

     1. In the principle value branches, the following formulae holds:

     sin^-1 (sin x) = x, cos^-1 (cos x) = x, tan^-1 (tan x) = x, cos^-1 (cot x) = x, sec^-1 (secx) = x, cosec^-1 (cosecx) = x

     2. If a is some constant angle, then

     -    sin q = sin a Þ q = np + (-1)ⁿ a, n Î Z  

     -    cos q = cos a Þ q = 2np ± a, n Î Z

     -    tan q = tan a Þ q = np + a, n Î Z