Trigonometry and Mensuration

Course- Diploma Semester-II

Subject- Advanced Mathematics Unit- III

RAI UNIVERSITY, AHMEDABAD

Unit-III Trigonometry and Mensuration 3.1 Trigonometry 3.1.1 Trigonometric Identities—

1. sin 휃 + cos 휃 = 1 2. sec 휃 − tan 휃 = 1 3. cosec 휃 − cot 휃 = 1

3.1.2 Compound angles formulas— 1. 푠푖푛(퐴 + 퐵) = 푠푖푛 퐴 푐표푠 퐵+ 푐표푠 퐴 푠푖푛 퐵 2. 푠푖푛(퐴 − 퐵) = 푠푖푛 퐴 푐표푠 퐵− 푐표푠 퐴 푠푖푛 퐵 3. 푐표푠(퐴 + 퐵) = 푐표푠 퐴 푐표푠 퐵 − 푠푖푛 퐴 푠푖푛 퐵 4. 푐표푠(퐴 − 퐵) = 푐표푠 퐴 푐표푠 퐵 + 푠푖푛 퐴 푠푖푛 퐵 5. 푡푎푛(퐴 + 퐵) =

6. 푡푎푛(퐴 − 퐵) =

7. 푐표푡(퐴 + 퐵) =

8. 푐표푡(퐴 − 퐵) = 3.1.3 Double angle formulas—

1. 푠푖푛 2푥 = 2 푠푖푛 푥 푐표푠 푥 =

2. 푐표푠 2푥 = 푐표푠 푥 − 푠푖푛 푥 = 2 푐표푠 푥 − 1 = 1 − 2 푠푖푛 푥 =

3. 푡푎푛 2푥 =

4. 푐표푡 2푥 = 3.1.4 Sub multiple angle formulas— 1. 푠푖푛 푥 = 2 푠푖푛 푐표푠

=

2. 푐표푠 푥 = 푐표푠 − 푠푖푛

= 2 푐표푠 − 1

= 1 − 2 푠푖푛

=

3. 푡푎푛 푥 =

4. 푐표푡 푥 =

Unit-III Trigonometry and Mensuration Example— Find the values of 퐬퐢퐧 ퟐퟐ ퟏ°

ퟐ, 퐜퐨퐬 ퟐퟐ ퟏ°

ퟐ and 퐭퐚퐧 ퟐퟐ ퟏ°

ퟐ by using submultiple angle

formula. Solution— We know that— cos푥 = 1 − 2 푠푖푛

∴ 푠푖푛 =

∴ sin =

On putting 푥 = 45°

sin 22 ° = °

=

√ = √ √

= √

∴ 풔풊풏ퟐퟐ ퟏ°ퟐ

= ퟐ √ퟐퟐ

Now, taking a right angle triangle as shown in fig— By using Pythagoras theorem— AB + BC = AC BC = √AC − AB BC = 2 + √2 Hence,

퐜퐨퐬ퟐퟐퟏ°ퟐ

=퐁퐂퐀퐂

=ퟐ + √ퟐퟐ

tan 22 ° = = √

√= √

√

= √√

× √√

= √√

퐭퐚퐧 ퟐퟐ ퟏ°ퟐ

=√ퟐ − ퟏ

ퟐퟐퟏ°ퟐ

2− √2 2

A

B C

Unit-III Trigonometry and Mensuration Example— Find the values of 퐬퐢퐧 ퟏퟖ°, 퐜퐨퐬 ퟏퟖ° and 퐭퐚퐧 ퟏퟖ° by using sub multiple angle formula. Solution— We know that— sin 푥 = 2 sin cos On putting 푥 = 72° sin 72° = 2 sin 36° cos 36°

sin(90° − 18°) = 2(2 sin 18° cos 18°)(1 − 2 sin 18°) cos 18° = 4 푠푖푛 18° 푐표푠 18°(1 − 2 푠푖푛 18°) 1 = 4 sin 18° (1 − 2 sin 18°) 8 sin 18° − 4 sin 18° + 1 = 0 (2 sin 18° − 1)(4 sin 18° + 2 sin 18° − 1) = 0 but sin 18° ≠ , therefore—

4 sin 18° + 2 sin 18° − 1 = 0 ∴ sin 18° = √

퐬퐢퐧 ퟏퟖ° =√ퟓ − ퟏퟒ

Now, taking a right angle triangle as shown in fig— By using Pythagoras theorem— AB + BC = AC BC = √AC − AB

BC = 10 + 2√5 Hence,

퐜퐨퐬 ퟏퟖ° =퐁퐂퐀퐂

=ퟏퟎ + ퟐ√ퟓ

ퟒ

tan 18° = = √

√= √

√

= √√

× √√

= 1 − √

퐭퐚퐧 ퟏퟖ° = ퟏ − ퟐ√ퟓퟓ

Selected + sign, since 퐬퐢퐧ퟏퟖ° > 0

ퟏퟖ°

√5 − 1

4

B C

A

Unit-III Trigonometry and Mensuration 3.1.5 Properties of triangle— 1. sine rule—

2. Cosine rule—

3. Relation between area of triangle, angles and its sides—

Where, 2s = a + b + c ∆ = area of triangle R = radius of circum circle r = radius of incircle

Unit-III Trigonometry and Mensuration 4. Half angle formulae—

EXERCISE Question— Find the values of sin 72 , cos 72 and tan 72 . Question— Find the sine of the angles A,B and C of triangle ABC having sides 푎 = 3, 푏 = 4 and 푐 = 5. Question— Find the in-circle radius of the triangle having its side 5cm, 12cm and 13cm. Question— Find the circum radius of the circle inscribing the triangle having its sides equal to 8cm, 15cm and 17cm. Question— Find the area of the triangle having its sides 4cm, 4cm and 6cm.

Unit-III Trigonometry and Mensuration 3.2 Mensuration 3.2.1 Introduction—anything concerned with measuring, calculating and estimating lengths, areas and volumes, as well as the construction of objects, comes under the Mensuration. Therefore, units have an important role in Mensuration. Some standard units with their conversion are listed below—

Length Area Volume Weight 1 Km 1000 m 1 m 10000 cm 1 m 1000 litres 1 tonne 1000 Kg

1 m 100 cm 1 cm 100 mm 1 litre 1000 ml 1 Kg 1000 g

1cm 10 mm 1 cm 1 ml 3.2.2 Triangle—a triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B and C is denoted by ∆ABC . Shape of a triangle is shown below— A

B C Existence of a triangle—the triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. That sum can equal the length of the third side only in the case of a degenerate triangle, one with collinear vertices. It is not possible for that sum to be less than the length of the third side. There are two important parameter related to a triangle are— Area of Triangle— It depends upon the shape and size of the triangle. Perimeter— It is equal to the sum of the sides of the triangle.

Perimeter = AB + BC + CA Some important types of triangle are listed below— A

1. Right angle triangle— Square of any one side of triangle is equal to the sum of squares of other sides.

Here, AB + BC = AC and triangle is right angled at B. Perimeter = AB + BC + CA B C Area = (AB)(BC)

Note— For any ∆ABC, ∠퐀 + ∠퐁 + ∠퐂 = ퟏퟖퟎ°

Unit-III Trigonometry and Mensuration 2. Isosceles triangle—A triangle having two equal sides is called as isosceles triangle.

In this diagram side AB and AC are equals. Hence, ∠B and ∠C are equals. A

Perimeter = 2AB + BC Area = (BC)√4AB − BC

B C 3. Equilateral triangle— A triangle having all sides equal is called as equilateral triangle.

In this diagram side AB, BC and AC all are equal. Hence, ∠A , ∠B and ∠C are equals. A

Perimeter = 3AB Area = √ AB

B C Quadrilateral— a quadrilateral is a polygon with four sides and four vertices or corners. There are two types of quadrilateral— A

1. Planar quadrilateral—The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is— B D

C 2. Crossed quadrilateral— In a crossed quadrilateral, the four A

interior angles on either side of the crossing add up to 720°.

B C D Parallelogram— A four-sided polygon with two pairs of parallel and equal sides. The following is a parallelogram— 푨풓풆풂 = 푩풂풔풆 × 풉풆풊품풉풕 Rectangle— A rectangle is a parallelogram with 4 right angles. The following is a rectangle—

h

b

Unit-III Trigonometry and Mensuration Square—A square is a rectangle with 4 equal sides. The following is square—

Rhombus— A rhombus is a parallelogram with 4 equal sides. The following is a rhombus—

Trapezoid— A trapezoid is a quadrilateral with only one pair of parallel sides. The following are trapezoids— 1. Scalene trapezoid— A scalene trapezoid is a trapezoid with no equal sides. The following is a scalene trapezoid— 2. Right-angled trapezoid— A right-angled is a trapezoid with two right angles. The following is a right-angle trapezoid— 3. Isosceles trapezoid— In an isosceles trapezoid, non-parallel sides are equal. The following is an isosceles trapezoid—

Unit-III Trigonometry and Mensuration Circle—

Semicircle—

Sphere— Sphere is a locus of the points having fixed distance from a fixed point in three dimensional planes.

Cone— It is the solid figure bounded by a base in a plane and by a surface (called the lateral surface) formed by the locus of all straight line segments joining the apex to the perimeter of the base.

Volume of the sphere =43πr

Surface area of the sphere = 4πr

A right circular cone and an oblique circular cone

Right circular cone—

Unit-III Trigonometry and Mensuration Cylinder—A cylinder is defined more broadly as any ruled surface spanned by a one-parameter family of parallel lines. A cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively.

A right circular cylinder Cube—

Cuboids—

Right circular cylinder— Lateral surface area = 2πrh Total surface area = 2πrh + 2πr Volume of the cone = πr h

Unit-III Trigonometry and Mensuration

EXERCISE Question— Three sides of a triangle are AB=3 cm, BC=4cm and CA=5cm. Discuss about the type of triangle. Also find its area. Question— A triangle having two equal sides of length 12 cm and third side of length 9 cm. Find the area of the triangle. Question— Find the sides of an equilateral triangle having area 173 sq. cm. Question— Find the volume of a cone having base radius 5cm and height 12cm. Question— Find the volume of a cylinder having base radius and height both equals to 7cm.

Unit-III Trigonometry and Mensuration References— 1. en.wikipedia.org/wiki/Trigonometry 2. www.mathsisfun.com 3. https://www.khanacademy.org/math/trigonometry 4. www.sosmath.com 5. www.themathpage.com 6. www.clarku.edu 7. en.wikibooks.org/wiki/Trigonometry 8. www.cimt.plymouth.ac.uk 9. www.bbc.co.uk