Math 105 PLANE TRIGONOMETRY

Math 105 – PLANE TRIGONOMETRY

MATHEMATICS 105

Plane Trigonometry

Chapter I The Trigonometric Functions

Introduction

The word trigonometry literally means triangle measurement. It is concerned with the measurement of the parts, sides, and angles of a triangle. It further deals with six ratios that are determined by angles and directed line segments. The main purpose of the study of trigonometry is to solve problems involving triangles as what are found in astronomy, navigation and surveying.

With the development of trigonometry, trigonometric functions are associated with the lengths of the arcs on the unit circle. It is now defined as a branch of Mathematics which is concerned with the properties and applications of circular or trigonometric functions.

Plane Trigonometry, which is the concentration of this course, is restricted to the study of triangles lying in a plane.

Directed Line Segments

Two concepts that are associated with a line segment are distance and direction. The distance from one point to another, or the length of the line segment between the two points, is the number of times that an accepted unit can be laid off along the given segment.

When a line segment is measured with a definite sense or direction from one endpoint to the other, the segment is said to be directed line segment.

The distance between two distinct points on a directed line segment is called a directed distance.

A fundamental property of directed line is that if P1, P2 and P3 are any distinct points on the line, then the following relation holds:

On line L , we can assign an origin, a unit length and a positive direction. Figure 1, illustrates a line where the distance between 0 and 1 corresponds to the unit length, the arrow at the right indicates positive direction. This line is a one-dimensional coordinate system or a number line, where a one-to-one correspondence can be established.

-3 -2 -1 0 1 2 3

The directed distance from P1(x1) to P2(x2) regardless of the relative positions of 0, P1 and P2 is given by

The undirected distance between P1 and P2 is the absolute vale of the directed distance between them.

Exercises:

Find the directed distance from P1 to P2 given the following:

No. / P1 / P2 / No. / P1 / P2
1 / 6 / - 4 / 6 / 3.580 / 8.098
2 / k / 3k / 7 / ¾ / 5
3 / 2 ½ / 5 ¾ / 8 / 2.37 / -5.3
4 / 7 / -8 / 9 / 7 ½ / 9.3
5 / -b / a / 10 / -1.0007 / 2.083

The Pythagorean Theorem

Pythagoras of Samos, a Greek mathematician, is best known for the Pythagorean theorem, which bears his name and is credited with its discovery and proof. He is also known as "the father of numbers".

The Pythagorean theorem states that:

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs

The theorem can be written as an equation: c2 = a2 + b2 where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides or the legs.

Examples:

1.  Find the length of the missing side in the following:

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Assistant Professor I

Math 105 – PLANE TRIGONOMETRY

i.  a = 12 b = 16

Solution: c = ?

c2=a2+b2

c2=122+162

=144+256

=400

c=20

ii.  b = 18 c = 20

Solution: a = ?

c2=a2+b2

202=a2+182

400=a2+324

400-324=a2

76=a2

a=8.72

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Assistant Professor I

Math 105 – PLANE TRIGONOMETRY

2.  Find the length of a rectangular lot 14 m wide and with 50m diagonal path.

Solution: a = 14m, c = 50m, b = ?

c2=a2+b2

502=142+b2

2500=196+b2

2500-196=b2

2304=b2 b=48 m

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Assistant Professor I

Math 105 – PLANE TRIGONOMETRY

The Rectangular Coordinate System

Points in a plane are located using coordinate axes which consists two perpendicular lines X’OX (horizontal) and Y’OY (vertical) which intersect at the point O (refer to the figure at the right). The line X’OX is called the x-axis and the line Y’OY is called the y-axis. Together they are known as the Cartesian coordinate axes. The point of intersection is called the origin.

The coordinate axes divide the plane into four distinct regions called quadrants marked I, II, III, IV (labelled counter clockwise). In quadrant I, both x and y coordinates are positive (x > 0, y > 0), in quadrant II, x-coordinate is negative, y-coordinate is positive (x < 0, y > 0); in quadrant III, both are negative (x < 0, y < 0) and in quadrant IV, x-coordinate is positive, y-coordinate is negative, (x > 0, y < 0).

Examples:

1.  Plot the points whose coordinates are:

a.  P1 (5, 4)

b.  P2 (4, -1)

c.  P3 (0, 3)

d.  P4 (-2, -5)

2.  Give the coordinates of the following points and the quadrants where they are located:

a.  five units to the right of the y-axis and two units below the x-axis

b.  two units to the left of the y-axis and one unit below the x-axis

The Distance Formula

The distance between two points P1 and P2 can be expressed in terms of their coordinates by the Pythagorean theorem:

Let the coordinates of two points be denoted by P1(x1, y1) and P2(x2, y2), by Pythagorean theorem,

Examples:

1.  Find the distance between two points:

a.  P1(5, 14) P2(-10, 2)

b.  P1(2, 5) P2(-1, 3)

If the two points in the distance formula are the origin and the point whose radius vector r we want, we find that r=x-02+y-02. Consequently, squaring each member of this equation, a relation between the coordinates and the radius vector r of P(x,y) is x2+y2=r2.

Examples:

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Assistant Professor I

Math 105 – PLANE TRIGONOMETRY

1.  If the abscissa of a point is 12 and its radius vector is 13, find the values of the ordinate.

Solution: x = 12, r = 13, y = ?

x2+y2=r2

122+y2=132

144+y2=169

y2=169-144

y2=25

y=±5

2.  If the radius vector of a point in the second quadrant is 5 and the ordinate is 3, find the abscissa.

Solution: x = ?, r = 5, y = 3

x2+y2=r2

x2+32=52

x2+9=25

y2=16

y=±4

Since the point is in the second quadrant,

we will use x = - 4

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Assistant Professor I

Math 105 – PLANE TRIGONOMETRY

Exercises:

1.  Plot each point whose coordinates are given and find the radius vector of each. Furthermore, find the distance between each pair of points.

A(3, 2) B(7, - 5) C(-8, 4) D(- 3, -6)

2.  Find the value or values of the one of x, y and r that is missing in each of the following:

a.  x = 3, y = 4

b.  x = -24 r = 25

c.  x = 5 y = 11

d.  x = -3 r = 5, y < 0

e.  y = -2 r = 27 x < 0

Angles

Angles are what trigonometry is all about. This is where it all started, way back when. Early astronomers needed a measure to tell something meaningful about the sun and moon and stars and their relationship between man standing on the earth or how they are positioned in relation to one another. Angles are the input values for trigonometric functions.

An angle is formed where two rays (straight line with an endpoint that extends infinitely in one direction) have a common endpoint. This endpoint is called the vertex. The two rays are called the sides of an angle – initial and terminal sides. A plane angle is to be thought of as generated by a revolving (in a plane) a ray from the initial position to a terminal position.

In Figure 1, ÐNOP or Ðq, ray OP OP is the initial side, while ray ON ON is the terminal side.

Writing Angle Names Correctly. An angle can be identified in several different ways:

·  Use the letter at the vertex of the angle.

·  Use the three letters that label the points – one on one side, the vertex and the last on the other ray. Points are labelled with capital letters.

·  Use the letter or number in the inside of the angle. Usually, the letters used are Greek or lowercase.

Examples:

Give all the different names that can be used to identify the angle shown in Figure 2.

Measure of an angle. An angle, so generated, is called positive if the direction of rotation is counterclockwise and negative if the direction of rotation is clockwise. The common unit of measure of an angle is degree denoted by ( ° ).

Classification of Angles. Angles can be classified by their size.

Ø  Acute Angle – an angle measuring less than 90°.

Ø  Right Angle – an angle measuring exactly 90°; the two sides are perpendicular

Ø  Obtuse Angle – an angle measuring greater than 90° and less than 180°.

Ø  Straight Angle – angle measuring exactly 180°.

Two angles can also be classified according to the sum of their measures. If the sum of the measures of the angles is 90°, then the angles are called complimentary angles. If the sum is 180°, then the angles are called supplementary angles.

When two line cross one another, four angles are formed. These angles which are opposite one another are called vertical angles. If two angles are vertical, then their measures are equal.

Examples:

1.  If one angle in a pair of supplementary angles measure 80°, what does the other angle measure?

Answer: The other measures 180° - 80° = 100°

2.  What is the measure of the (a) complement (b) supplement of the angle in Figure 1 if q is 38°?

Answer: (a) complement is 90° - 38° = 52°

(b) supplement is 180° - 38° = 142°

Exercises:

Prepared by Mrs. Koni Gutierrez Cruz | Bataan Peninsula State University

Assistant Professor I

Math 105 – PLANE TRIGONOMETRY

1.  Find the measure of the complement of an angle whose measure is:

a.  37°

b.  58°

c.  x°

d.  90° - y°

e.  37° + R°

2.  Determine the measure of the supplement of an angle whose measure is:

a.  75°

b.  85°

c.  x°

d.  90° + y°

e.  113.76° + R°

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Assistant Professor I

Math 105 – PLANE TRIGONOMETRY

Angle in Standard Position

An angle is in standard position with reference to a system of rectangular coordinate axes if its vertex is at the origin and its initial side lines along the positive ray of the X axis.

If an angle is in standard position, the angle is said to be in the quadrant in which the terminal side lies. Hence, an acute angle is in the first quadrant; an obtuse angle is in the second quadrant; an angle of 215° is in the third quadrant; an angle of 330° is in the fourth quadrant.

If the terminal side of an angle in standard position coincides with one of the coordinate axes, the angle is a quadrantal angle. An angle of 90° and any angle which is an integral multiple of 90° is a quadrantal angle. Two angles are coterminal if they are in standard position and have the same terminal sides. Thus, of 150°, 510°, and -210° are coterminal angles.

Exercises:

1.  Construct the following angles in standard position and determine those which are coterminal:

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Assistant Professor I

Math 105 – PLANE TRIGONOMETRY

a.  125°

b.  210°

c.  -150°

d.  385°

e.  930°

f.  -370°

g.  -955°

h.  -870°

i.  595°

j.  -210°

Prepared by Mrs. Koni Gutierrez Cruz | Bataan Peninsula State University

Assistant Professor I

Math 105 – PLANE TRIGONOMETRY

2.  Give three other angles coterminal with

Prepared by Mrs. Koni Gutierrez Cruz | Bataan Peninsula State University

Assistant Professor I

Math 105 – PLANE TRIGONOMETRY

a.  125°.

b.  45°

c.  165°

d.  590°

e.  -30°

f.  75°

g.  -120°

h.  -60°

i.  270°

j.  135°

Prepared by Mrs. Koni Gutierrez Cruz | Bataan Peninsula State University

Assistant Professor I

Math 105 – PLANE TRIGONOMETRY

Prepared by Mrs. Koni Gutierrez Cruz | Bataan Peninsula State University

Assistant Professor I

Math 105 – PLANE TRIGONOMETRY

Definition of the Trigonometric Functions of Any Angle in Standard Position

The six ratios on which the subject of trigonometry is based are formed by using the abscissa, the ordinate, and the radius vector of a point on the terminal side of an angle in standard position. Since the values of these ratios depend on the values of the abscissa, the ordinate, and the radius vector, and these in turn depend on the size of the angle, each ratio is a function of the angle.

We shall now give the definitions of these six functions and, beside each definition, the abbreviated form in which it is usually written. These definitions and abbreviated forms should be memorized. If the angle q is in standard position, then