23 Maths -- Trigonometry

ask mattrab Visit www.askmattrab.com for more academic resources.

The Projection Law

0 Questions 2305 Views

From the Sine Law, We have,
b =2R sin B...........(i)
c =2R sin C...........(ii)
=2R sin A...........(iii)

To Prove: i) b cos C + c cos B =
Taking LHS;
= b cos C + c cos B
= 2R sin B. cos C + 2R cos B. sin C  [from eqn (i) & (ii)]
=2R(sin B. cos C + cos B. sin C )
= 2R sin(B + C)
= 2R sin A   [B + C=180^O - A, so, sin(B + C)=sin(180^O - A)= sin A]
== R.H.S.   [from eqn (iii)]

To Prove: ii) c cos A +  cos C = b
Taking LHS;
= c cos A +  cos C
= 2R sin C. cos A + 2R cos C. sin A  [from eqn (ii) & (iii)]
= 2R(sin C. cos A + cos C. sin A)
= 2R sin(C + A)
= 2R sin B   [ C + A=180^O - B, so, sin(C + A)=sin(180^O - B)= sin B]
= b = R.H.S.     [from eqn (i)]

To Prove: iii) cos B + b cos A = c
Taking LHS;
= cos B + b cos A
= 2R sin A. cos B + 2R cos A. sin B  [from eqn (iii) & (i)]
= 2R(sin A. cos B + cos A. sin B)
= 2R sin(A + B)
= 2R sin C   [ A + B=180^O - C, so, sin(A + B)=sin(180^O - C)= sin C]
= c = R.H.S.     [from eqn (ii)]

More notes on Trigonometry

Introduction to Trigonometry

Important Points For Solution of Triangle

⁕ For Oblique Triangle:1. When three angles of a triangle are given, we use Sine Law to find the ratio of the sides of the triangle i.e. 2. When two angles and one side of a ...

Solutions of triangle

Solution of triangle ( introduction and points to be noted)A triangle has six elements, three angles and three sides. Out of these six, if any three of them are known then the process of determinin...

Trigonometric Functions

Solutions to Trigonometric Equations

Theorem 1: The Sine Law

⁕ Statement: In any triangle the sides are proportional to the Sine of the opposite angles.In other words, in any ΔABC, Proof:↪ Let ABC be a triangle with a = BC, b = CA and c = AB, then three cas...

Theorem 2: The Sine Law in terms of Circum-Radius R

⁕ Statement: In any ΔABC, where R is the Circum-Radius of ΔABC.Proof:↪ Consider the circum-circle of ΔABC with center O and radius R. Then we can have three possible figures.Where the angle A is a...

Properties of Triangle - Formula

Introduction:A triangle consists of three side and three angles which is collectively known as the six elements. In a triangle ABC, the side opposite to ∠A is denoted by '', the side opposite to ...

The Cosine Law

Proof:↪ We consider a triangle ABC then we can have three possible figures where angle C is acute in figure i, right angle in figure ii and obtuse in figure iii.From A draw AD perpendicular to BC (...

The Sine Law

Let ABC be a triangle, acute angled in fig(i), obtuse angled in fig(ii) and right angled in fig(iii).A perpendicular AD is drawn on BC [BC is produced to D in fig. (ii) and D coincides C in fig (ii...

The Cosine Law - Proof

Let ABC be a triangle, acute angled in fig. (i), obtuse angled in fig. (ii) and right angled in fig (iii) .BD is drawn perpendicular to AC from point B in fig(i) and fig(ii).To prove: Proof:1. Whe...

The Sine Law in terms of Circum-Radius

Let ABC be the triangle with all three vertex on the circumference of the circle with center O as shown in fig above.BO is drawn if the angle is either acute or obtuse as shown in fig(i) and fig(ii...