## Wednesday, April 11, 2018

### Think, Believe, Dream and Dare

Think, Believe, Dream and Dare

An eight-year-old boy approached an old man in front of a wishing well, looked up into his eyes, and asked: "I understand you're a very wise man. I'd like to know the secret of life."

The old man looked down at the youngster and replied: "I've thought a lot in my lifetime, and the secret can be summed up in four words.

The first is think. Think about the values you wish to live your life by.

The second is believe. Believe in yourself based on the thinking you've done about the values you're going to live your life by.

The third is dream. Dream about the things that can be, based on your belief in yourself and the values you're going to live by.

The last is dare. Dare to make your dreams become reality, based on your belief in yourself and your values."

And with that, Walter E. Disney said to the little boy, "Think, Believe, Dream, and Dare."

~ Author Unknown ~

from: Harmony Chain Central Secretariat

## Thursday, June 25, 2015

LESSONS  IN  MATHEMATICS ..... CLICK YOUR PREFERENCE

4.  TRIGONOMETRY ( LESSON 1 )

4.a.  TRIGONOMETRY ( LESSON 2 -  6 )

7.  CIRCLES

8.  PARABOLA

10.  SETS

12.  DEPRECIATION, DEPLETION AND VALUATION OF PROPERTIES

19.  PHYSICS

20.  SPANISH

## Saturday, June 20, 2015

### Trigonometry : Lesson 1

TRIGONOMETRY

1. Trigonometry – is a branch of mathematics that deals with the relationship between the angles and sides of a triangle and the theory of the periodic functions connected with them.

2.  Triangle – a figure formed by three line segments joining three points that are not in the same plane.

3.  Right triangle – a triangle with at least one right angle.

4.  Oblique triangle – any triangle which has no right angle.
Oblique triangles are classified according to sides and according to angles as :
i) According to sides                   ii)  According to angles
a) equilateral triangle                   a)  equiangular triangle
b) isosceles  triangle                    b)  acute triangle
c) scalene triangle                       c)  obtuse triangle

5.  Congruent triangles – two triangles with the same shape and size.

6.  Similar triangles – two triangles with the same shape but not necessarily of the same size.

7. Altitude of a triangle – the length of the line segment from any vertex of a triangle that is  perpendicular to the opposite side.

8.  Hypotenuse – the longest side of the triangle that is opposite to the right angle.

9. Perimeter of a triangle – is the distance around the triangle and is equal to the sum of the three sides.

10. The area of a triangle is ½  of the base ( b ) times the altitude ( a ).

A = ½ b a

If the three sides are given, the area is given by the formula
where  s =  ½ ( a + b + c )

RIGHT TRIANGLES

Figure 1

11. Pythagorean theorem – in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

c2  =  a2  +  b2

b2  =  c2 a2

a2  =  c2 b2

*  Proofs of Pythagorean theorem

( as presented in the classroom discussion )

Examples :  Find the missing side, the perimeter and the area of the given right triangle.

1.  a =  12 cm ,  c =  13  cm                      7.   a  =  24 cm ,    c  =  40 cm

2.  b = 12 cm ,  a = 9 cm                           8.   b  =  36 cm ,    c  =  45 cm

3.  c = 35 cm,   b = 21 cm                         9.   a  =  33 cm ,    b  =  44 cm

4.  a = 36 cm,   b = 48 cm                        10.  b  =  48 cm ,     c  =  80 cm

5.  a = 81 cm,   c =  135 cm                     11.  a  =  63 cm ,     c  =  105 cm

6.  c = 12 cm,   a =  6 cm                         12.  a  =  78  cm ,     b  =  104  cm

13.  Will a round glass table top, 2.5 m in diameter, fit through a doorway which is 2.13 m
high  and 0.92 m  wide ?

14. A rectangle has a base of 12 cm. If the diagonal is 15 cm, determine the altitude of the rectangle.

15.  The area of a square is 128 cm2. Determine the length of its diagonal.

From Geometry :

The sum of the measure of the three angles of a triangle is equal to  180°

From the figure above :

a +  b  +  90°  =  180°

a +  b =  180° – 90°

a +  b = 90°

Hence, the sum of the measure of the acute angles of a right triangle is equal to 90°.

Example :  Find   a if  b  = 39° 28’.

Solution :    a  =  90° – 39° 28’  =  50°  32’

Exercises :  Find   a  or  b  if

1.  a = 36° 24’  32’’                        6.  b = 56°  43’  27’’

2.  b =  68°  13’                             7.  a= 62° 54’  12’’

3.   a =  47° 19’                             8.   b = 73° 24’  32’’

4.  a =  17°  21’  36”                      9.  a = 16° 48’

5.  b =  67°  34’  26”                     10.  b  = 22° 47’

PROPERTIES OF SIMILAR TRIANGLES

Basic principle :
The ratio of any two sides of a right triangle with an acute angle depends only on the size of the angle and not on the size of the triangle.
Memory aid
SOH                    CHO
CAH                    SHA
TOA                    CAO

The six trigonometric ratios are called trigonometric functions.

The value of each of the six trigonometric functions of an acute  angle  is determined  when  the acute angle is given.  Furthermore, it can be shown that, if the value of one of the six trigonometric functions of an acute angle is equal to the value of the same function of a second acute angle, the two acute angles are equal.

Example
1.    OA = 9 cm,  AX = 12 cm,   OX =  15 cm
OB = 21 cm,  BY = 28 cm,   OY =  35 cm
OC = 33 cm,  CZ = 44 cm,   OZ =  55 cm

2.   OA = 5 cm,  AX = 12 cm,   OX =  13 cm
OB = 10 cm,  BY = 24 cm,   OY =  26 cm
OC = 20 cm,  CZ = 48 cm,   OZ =  52 cm

3.   OA = 6 cm,  AX = 8 cm,   OX =  10 cm
OB = 12 cm,  BY = 16 cm,   OY =  20 cm
OC = 24 cm,  CZ = 32 cm,   OZ =  40 cm

FUNCTIONS OF ACUTE ANGLE

1.  Sin a                      4.  csc a
2.  cos a                      5.  sec a
3.  tan a                       6.  Cot a

Exercises

1.  If sin a =  3/5, find cos a, tan a,  sec a,  csc a  and  cot a.

2.  If tan a =  5/12, find cos  a, sin  a,  sec a,  csc  a  and  cot  a.

3.  If cot a =  8/15, find cos  a, tan  a,  sec a,  csc  a  and  sin  a.

4.  If cos a  =  12/13, find sin  a, tan  a,  sec  a,  csc  a  and  cot  a.

5.  If csc a =  2, find cos a, tan a,  sec a,  sin a  and  cot a.

6.  If sec  a =  5/4, find cos a, tan a,  sin  a,  csc  a  and  cot  a.

7.  If cos  a =  4/5, find sin  a, tan  a,  sec  a,  csc  a  and  cot  a.

8.  If tan  a =  5/12, find cos  a, sin  a,  sec  a,  csc  a  and  cot  a.

9.  Given a right triangle
Prove that    a)  sin a/cos a  =  tan a

b)  sin a cos a  =  ab /( a2 + b2 )

c)  ( sin  a ) 2  +  ( cos  a ) 2   =  1

d)  ( sec  a ) 2  –  ( tan  a ) 2   =  1

e)  ( csc  a ) 2  –  ( cot  a ) 2   =   1

10. In the figure, RS = 60 cm, cos a = 4/5, sin  = 5/13,  determine the length of the sides
QR, QS and  QP.
11.  Using the same figure, determine the unknowns if  sin a = 3/5, and cos O = 12/13 .

12.  Two guy wires are attached to a pole 12 m above the ground level. They make angles
of  43° and  72°  with the ground at points which are in a line with the base of the pole.
How long are the wires ?

13.  At low tide the angle of elevation to the top of  12 m tall tree from the water’s edge is
6° 30'. At high tide the angle of elevation to the top of the tree is  6°. How high does the          water  level rises during the high tide along the line perpendicular to the shore ?

14. Determine the height of a tower if from a window in a building 95.735 m away the angle
of depression to the base of the tower is 68° while the angle of elevation to the top is 62°.

15.  A light house casts a shadow of 9 m when the angle of elevation of the sun is 56°. How
high is the light house?

16. The angle of elevation to a balloon from a point on the ground is 32°.  After a vertical
ascent of  68 m the angle of depression from the balloon to the same point on the ground       is 47°. What are the original and present heights of the balloon ?

17. From a point 25 m from the base of a tower, a bird flies to its top in a straight path at an
angle of elevation of 75° 32’.  How long is the flight path of the bird ? What is the height
of the tower ?
sin a   =  a/c                      cos b = a /c

cos a   =  b/c                     sin b = b /c

tan a   =  a /b                    cot b = a /b

sec a   = c /b                     csc b = c /b

csc a   =  c / a                   sec b = c / a

cot a  =  b / a                    tan b =  b / a

sin a  =  cos b  =  a / c

cos a  =  sin b  =  b / c

tan a  =  cot b  =  a / b

sec a  =  csc b  =  c / b

csc a  =  sec b  =  c / a

cot a  =  tan b  =  b / a

FUNCTIONS AND CO-FUNCTIONS

Sine  ---  cosine

Cosine ----  sine

Tangent --- cotangent

Cotangent ---  tangent

Secant ---  cosecant

Cosecant ---  secant

RECIPROCAL  FUNCTIONS

sin a = 1/ csc a

cos a  =  1/ sec a

tan a  =  1/ cot a

csc a  =  1/ sin a

sec a  =  1/ cos a

cot a  =  1/ tan a

Any function of a is equal to the co-function of the complement of a ( B ).

Examples :
1.  sin  30°  =   cos 60°  =  0.5

2.  tan  15°  =  cot  75°  =  0.26794919

3.   sec 36°  =  csc  54°  =  1.236067977

4.  cos  25°  =  sin  65°  =  0.906307787

SOLUTION OF RIGHT TRIANGLES
Formulas :

1.  a2 +  b2  = c2

2.  a  +  b  = 90°

3.  sin a  =  a / c  =  cos b

4.  cos a  =  b / c  =  sin b

5.  tan a  =  a / b  =  cot b

6.  cot a  =  b / a  =  tan b

7.  sec a  =  c / b  =  csc b

8.  csc a  =  c / a  =  sec b

Example :
Find the acute angles of the triangle whose base is 15 cm and an altitude of 20 cm.

From the figure above,  tan a =  20/15  =  1.333333333

a = Arc tan 1.3333333  =  53.13°

tan b =  b/a  =  15/ 20  =  0.75

b  =  Arct tan 0.75  =  36.87°

Exercises : Find the measure of the angle a and B of the following triangles using
calculator.

1.   a =  18,   b =  25                            6.  a =  21,  c =  40

2.  a = 40,  b = 60                                7.  b =  50,  c =  100

3.  b = 17.321,  c = 20                         8.  a =  12,  c =  15

4.  a  =  12 ,  b =  24                           9.  a  =  15,  b = 28

5.  b  =  18,   c =  30                         10.  b  =  35,   c =  44

SUGGESTIONS FOR SOLVING RIGHT TRIANGLES.

1. Make a preliminary sketch roughly to scale for the given data.

2.  To find any unknown part, use a formula which involves it but no other unknown.

3.  Check the result by substituting it in any of the equations from  1 to 8.

ANGLE OF ELEVATION AND DEPRESSION

*  Angle of elevation – the angle formed by a horizontal ray  and the observer’s “line of sight” to any point above the horizontal.

*  Angle of depression – the angle formed by a horizontal ray and the observer’s “line of
sight” to  any point below the horizontal.

Examples :

1.  A 120 m high tower casts a shadow of 60 m. Determine the angle of elevation of the sun.

2.  A light house is 14 m high. Determine the length of its shadow when angle of elevation of the sun is 52°.

3.  From an airplane flying at 2,135 m high above the ground, the angle of depression of a
landing field is 19° 32’. Determine the line of sight distance and the ground distance from
the plane to the field. (The ground distance is the distance of the plane from the field to the  point on the ground under the plane. )

4.  Determine the height of a tower if the angle of elevation of its top is  56° 35’ when seen from a point 206 m from the base of the tower.

5.  In flying upward  for  1,152 m along  a straight inclined path, an airplane rises  180 m.
Determine the climbing angle of the plane. ( Climbing angle is the angle of inclination of the plane from the horizontal ).

6.  From a ground distance of 1,100 m,  an airplane starts a straight glide for the edge of an airfield at a gliding angle of  16° from the horizontal. From what altitude did the glide starts ?

End of Lesson 1