Lesson Notes By Weeks and Term - Senior Secondary School 1

Application of Sine, Cosine and Tangent

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SUBJECT: MATHEMATICS

CLASS:  SS 1

DATE:

TERM: 2nd TERM

REFERENCE BOOK

  • New General Mathematics SSS 1 M.F. Macrae et al 
  • WABP Essential Mathematics For Senior Secondary Schools 1 A.J.S Oluwasanmi

 

 
WEEK EIGHT

TOPIC: Application of Sine, Cosine and Tangent

  • Application of sine, cosine and tangent, simple problems with respect to right angle triangles. 
  • Angles of elevation and depression 
  • Bearing and distances of places strictly by application of trigonometric ratio. 

(a)









In the figure above,triangle ABC is any triangle, right-angled at A.

tan B = b/c ,   tan C = c/b            (tan:opp/adj.)

sin B = b/a  ,   sin C = c/a             (sin:opp/hyp.)

Cos B = c/a  ,   cos C = b/a           (cos:adj./hyp.)

In ABC, B and C are complementary angles(i.e. B + C = 90).

 

If B = Ѳ, then C= 90o – Ѳ (as in below)









Sin Ѳ = Cos (90o – Ѳ) = b/a    and

CosѲ = sin (90o – Ѳ) = c/a

 

Examples

1)Calculate a)/PR/, b)/RS/ in the figure below.Give the answers correct to 3 s.f.







Solution

a)In triangle PQR,

tan 50o = x/6

x = 6 X tan 50o= 6 X 1.192

                       = 7.152

/PR/ = 7.15 cm to 3 s.f.

  1. In triangle PRS, 

Sin 28o = y/x

y = x sin 28 = 7.152 x 0.4695 = 3.358

/RS/ = 3.36 cm to 3 s.f.

2) A ladder of length 6.0 m rests with its foot on a horizontal ground and leans against a vertical wall.The inclination of the ladder to the horizontal is 80o. Find correct to one decimal place

  1. a) the distance of the foot of the ladder from the wall, b) the height above the ground at which the upper  end of the ladder touches  the wall. 

 

Solution







In the figure above,

       AC is the ladder   

       MB is the wall

       ACB is the inclination of the ladder to the horizontal = 80o

  1. The distance of the foot of the ladder from the wall is BC, where

Cos 80o = BC/6

BC = 6 Cos 80o = 6 X 0.1736 = 1.0416 m

                         = 1.0 m to 1 d.p.

  1. The height above the ground at which the upper end of the ladder touches the wall is AB, where

Sin 80o = AB/6

AB = 6 Sin 80o = 6 x 0.9848 = 5.9088 m

       = 5.9 m to 1 d.p.

 

 

  • Angle of elevation and depression 

 

The figure below shows the angle of elevation e , of the top of the tower, R , from a point A below. The diagram also shows the angle of depression, d , of a point B on the ground from a point, p , on the tower.












Examples

1)From a window 10m above level ground , the angle of depression of an object on the ground is 25.4o.Calculate the distance of the object from the foot of the building.

Solution








tan 25.4o = 10/d

d = 10/tan 25.4o

    = 10/0.4748

    = 21.06m

The object is 21.06m from the foot of the building

 

Bearing and Distances        

Examples

1)The bearing of X from Y is 046o. What is the bearing of Y from X ?

Solution








The bearing of X from Y is XYP = 046O

The bearing of Y from X is reflex   XY = Ѳ

Ѳ =180 + 46 = 226o

 

EVALUATION

1)The angle of elevation from the top of a tower from a point on the horizontal ground, 40m away from the foot of the tower , is 30o. Calculate the height of the tower to two significant figures.

2)From the top of a light-tower 40m above sea level, a ship is observed at an angle of depression of 6o. Calculate the distance of the ship from the foot of the light-tower, correct to 2s.f.

3)From a point P, R is 8km due east and 8km due south. Find the bearing of P from R

 

GENERAL EVALUATION 

  1. Express the sine, cosine and tangent of (a) 300, (b) 1500, (c) 2100, (d) 3300 as either a positive or a negative trigonometrical ratio of an acute angle. 

 

READING ASSIGNMENT

NGM BK 1 PG 114 – 129; Ex 11e nos 1 - 10

 

WEEKEND ASSIGNMENT

1)A town Y is 200 Km from town X in a direction 040o. How far is Y east of X?

a)125.8km    b)128.6km   c)127km   d)126.8km

2)A boy walks 1260m on a bearing of 120o. How far Southis he from his starting point?

a)630m    b)530m   c)730m   d)630km

 

Use the figures below to answer questions 3-5







Calculate to 2 s.f., the values of 

3)w    a)0.21    b)21    c)2.1    d)2.31

4)x      a)5         b)8       c)9       d)10  

5)y      a)6.5      b)7.5    c)8.5    d)9.5

THEORY

  1. A rhombus has sides 11 cm long. The shorter diagonal of the rhombus is 8cm long. Find the size of one of the smaller angles of the rhombus correct to the nearest degree.
  2. a)Fron the top of a cliff, the angle of depression of a boat on the sea is 22o.If the height height of the cliff above sea level is 40m, calculate, correct to 2 significant figures, the distance of the boat from the bottom of the cliff.

b)At a point 20m from the base of a water tank, the angle of elevation of the top of the tank is 45o. What is the height of the tank?





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