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Top Scorers for FINALS

Given on March 22 – 24, 2017

PHYSICS E 22 – Modern Physics

Name Score

1. Geisler Cabayao 96

2. Korina T. Dayot 94

3. Nathalie Gay C. Patricio 93

4. Carmel Therese Triumfante 92

5. Amerah Ampatuan 91

6. Kenneth Gallano 90

7. Aushe Mae Montebon 86

8. Cristy Juliet Reyes 82

PHYS 21 Fundamentals of Physics

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1. Farouk Gumaan 96

2. Russel Ann Completano 95

3. Bai Astrafiah Cassandra Abas 92

Krizza Joy Buckles 92

4. Baialmanisan Dimasangkay 88

MATH 23/ MATH 201 Plane Trigonometry

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1. Melanie Talusan 98

2. Angelo Florague 84

3. Mhil Rose Zhine Taytay 82

4. Benjie Ibot 80

MATH 25 Analytic Geometry

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1. Melanie Talusan 100

2. Benjie Ibot 92

3. Ivy Valenzuela 90

Johwena Cuevas 90

4. Angelo Florague 80

MATH 11 College Algebra

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1. Kevin Guiang 88

Wahida Manial 88

2. Winona Dojinog 85

3. Kathleen Faye Dacanay 83

MATH 22C Automata & Language Theory

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1. Ritche Sal grande 98

2. Jay Harley Gadingan 94

Cristy Joy Ybanez 94

3. Charlotte Evangelio 92

4. Joevin Niel Poquita 91

5. Jeanifer Aban 86

6. Kenneth Jude Carrera 83

7. Vicio Untal III 81

LANG 32 – Foreign Language 2

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1. Jane Mae Tampus 100

Al Suod Mascud 100

Mohamad Mustapha 100

Kimberly Olmoguez 100

Wahida Manial 100

Kapia Sabpa 100

2. Benny Chua 98

Samraida Mamay 98

Alchelou Balogo 98

3. Adrian Camatac 96

Ervin Ibot 96

Bainisan Balad 96

4. Sheena Rose Eduarte 84

Efren F. Cadungog, Sr., MIM

Class adviser

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.

P = a + b + c

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

RIGHT TRIANGLES

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

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

1. a = 12 cm , c = 13 cm

2. b = 12 cm , a = 9 cm

3. c = 35 cm, b = 21 cm

4. a = 36 cm, b = 48 cm

5. a = 81 cm, c = 135 cm

6. c = 12 cm, a = 6 cm

7. Will a round glass table top, 2.5 m in diameter, fit through a doorway

which is 2.13 m high and 0.96 m wide ?

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

the altitude of the rectangle.

9. 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°.

Examples : Find a or b.

1. a = 36° 24’ 32’’ 5. b = 56° 43’ 27’’

2. b = 68° 13’ 6. a = 62° 54’ 12’’

3. a = 47° 19’ 7. b = 73° 24’ 32’’

4. a = 17° 21’ 36” 8. a = 16° 48’

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.

Ratios

AX = BY = CZ = opposite side = sine a = sin a .

OX OY OZ hypotenuse

OA = OB = OC = adjacent side = cosine a = cos a .

OX OY OZ hypotenuse

AX = BY = CZ = opposite side = tangent a = tan a .

OA OB OC adjacent side

OA = OB = OC = adjacent side = cotangent a = cot a .

AX BY CZ opposite side

OX = OY = OZ = hypotenuse = secant a = sec a .

OA OB OC adjacent side

OX = OY = OZ = hypotenuse = cosecant a = csc a .

AX BY CZ opposite side

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 q = 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 q = 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 60°. At high tide the angle of elevation to the top of the tree is 30°.

How high does the water level rises during the high tide along the line

perpendicular to the shore ?

14. Determine the height of the World Trade Center in New York City if from a window

in a building 91.5 m away the angle of depression to the base of the Trade Center

is 68° while the angle of elevation to the top is 62°.

15. A light house casts a shadow of 9m 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 ?

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TOP SCORERS FOR FINALS GIVEN LAST MARCH 22 - 25, 2017

Friday, December 9, 2016

Lesson 1 Trigonometry