33. Elementary trigonometry

New Chapter

34. Solution of triangles and other applications of trigonometry

Study Plan

October

**22**

Study sections 34.1 to 34.5

Revision points

1. Semiperimeter of a triangle is denoted by s.

2. Area of a triangle is denoted by Δ or S.

3. a,b, and c represent sides BC,CA, and AB

4. Sine rule

In any Δ ABC

Sin A/a = Sin B/b = Sin C/c

5. Cosine Formulae

In any Δ ABC

Cos A = [b² + c² -a²]/2bc

Cos B = [c² +a² –b²]/2ac

Cos C = [a² + b² –c²]/2ab

6. Projection formulae

In any Δ ABC

a = b Cos C + C cos B

b= c Cos A + A Cos C

c = a Cos B + b cos A

7. Trigonometrical ratios of half of the angles of a triangle

1. Sin A/2 = √[(s-b)(s-c)/bc]

2. Cos A/2 = √[s(s-a)/bc]

3. tan A/2 = √(s-b)(s-c)/s(s-a)]

**23 October**

Study Sections 34.6, 34.7, 34.8, 34.9,

Revision points

8. Area of a triangle

S = ½ ab Sin C = ½ bc sina = ½ ac sin B

9. Napier’s analogy

In any triangle ABC

Tan [(b-c)/2] = [(b-c)cot (A/2)]/(b+c)

10. Circumcircle of a triangle

The circle which passes through the angular points or vertices of a triangle ABC is called its circumcircle.

The centre of this circle can be found by locating the point of intersection of perpendicular bisectors of the sides. It is called circumcentre.

The circumcentre may lie within, outside or upon one of the sides of the triangle.

In a right angled triangle the cicumcentre is vertex where right angle is formed.

The radius of circumcircle is denoted by R.

R = a/(2 Sin A) = b/(2 sin B) = c/(2 sin C)

11. Inscribed circle or incircle of a triangle

It is the circle touches each of the sides of the triangle.

The centre of the inscribed circle is the point of intersection of bisectors of the angles of the triangle.

The radius of inscribed circle is denoted by r (it is called in-radius) and it is equal to the length of the perpendicular from its centre to any of the sides of the triangle.

Various formulas that give r.

In- radius ( r )= Δ/s

r = (s-a)tan (A/2) = (s-b) tan (B/2) = (s-c) tan (C/2)

r = [a sin B/2 sin C/2/(Cos A/2)

r = 4R sin (A/2) sin (B/2) sin (C/2)

Attempt Objective Type Exercises 1 to 10

**October 24th**

34.10, 34.11

Revision points

12. Escribed circles of a triangle

The circle which touches the sides BC and two sides AB and AC produced of a triangle ABC is called the escribed circle opposite to the angle A. Its radius is denoted by r1.

Similarly r2 and r3 denote the radii of the escribed circles opposite to the angles B and C respectively.

The centres of the escribed circles are called the ex-centres.

13. Orthocentre and its distances from the angular points of a triangle

In a Δ ABC, the point at which perpendiculars drawn from the three vertices (heights) meet, it called the ortho centre of the ΔABC

Attempt objective type exercises 11 to 20

October 25th

Study 34.12 to 34.15

Revision points

14. Regular polygon and Radii of the inscribed and circumscribing circles of a regular polygon

the centre of the polygon will be the in-centre as well as circumcentre of the polygon.

15. Area of a cyclic quadrilateral

a quadrilateral is a cyclic quadrilateral if its vertices lie on a circle.

Area of cyclic quadrilateral = ½ (ab + cd) sin B

16. Ptolemy’s theorem

In a cyclic quadrilateral ABCD, AC.BD = AB.CD + BC.AD

The product of diagonals is equal to the sum of the products of the lengths of opposite sides.

17. Circum-radius of a cyclic quadrilateral

In a cyclic quadrilateral, the circumcircle of the quadrilateral ABCD is also the circumcircle of Δ ABC.

Attempt obj type exercises 21 to 30.

October 26th

Attempt obj type exercises 31 to 45.

October 27th

Attempt obj type exercises 46 to 60.

October 28th

Attempt obj type exercises 61 to 75.

October 29th

Attempt obj type exercises 76 to 90.

October 30th

Attempt fill in the blanks 1 to 15.

October 31st

Attempt fill in the blanks 16 to 31.

Practice Exercise at the end of the chapter has 21 problems you have to do these problems during the next 10 days as a part of the revision of earlier chapter. 2 problems a day will take care of all the 21 problems.

Problems of Ch. 33. Elementary trigonometry whihc you have not done so far, have to done as revision part during these 10 days.

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