MAC 1114

1
MAC 1114

• Module 1
• Trigonometric Functions

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Learning Objectives

• Upon completing this module, you should be able
  to
• Use basic terms associated with angles.
• Find measures of complementary and supplementary
  angles.
• Calculate with degrees, minutes, and seconds.
• Convert between decimal degrees and degrees,
  minutes, and seconds.
• Identify the characteristics of an angle in
  standard position.
• Find measures of coterminal angles.
• Find angle measures by using geometric
  properties.
• Apply the angle sum of a triangle property.

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3
Learning Objectives (Cont.)

• Find angle measures and side lengths in similar
  triangles.
• Solve applications involving similar triangles.
• Learn basic concepts about trigonometric
  functions.
• Find function values of an angle or quadrantal
  angles.
• Decide whether a value is in the range of a
  trigonometric function
• Use the reciprocal, Pythagorean and quotient
  identities.
• Identify the quadrant of an angle.
• Find other function values given one value and
  the quadrant.

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Trigonometric Functions
There are four major topics in this module
- Angles - Angle Relationships and Similar
Triangles - Trigonometric Functions - Using the
Definitions of the Trigonometric Functions
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What are the basic terms?

• Two distinct points determine a line called line
  AB.
• Line segment ABa portion of the line between A
  and B, including points A and B.
• Ray ABportion of line AB that starts at A and
  continues through B, and on past B.

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What are the basic terms? (cont.)

• Angle-formed by rotating a ray around its
  endpoint.
• The ray in its initial position is called the
  initial side of the angle.
• The ray in its location after the rotation is the
  terminal side of the angle.

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How to Identify a Positive Angle and a Negative
Angle?

• Negative angle The rotation of the terminal side
  is clockwise.

• Positive angle The rotation of the terminal side
  of an angle counterclockwise.

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Most Common unit and Types of Angles

• The most common unit for measuring angles is the
  degree.
• The major types of angles are acute angle, right
  angle, obtuse angle and straight angle.

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What are Complementary Angles?

• When the two angles form a right angle, they are
  complementary angles. Thus, we can find the
  measure of each angle in this case.

The two angles have measures of 43 20 63
and 43 - 16 27
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What are Supplementary Angles?

• When the two angles form a straightangle, they
  are supplementary angles. Thus, we can find the
  measure of each angle in this case too.

These angle measures are 6(19) 7 121 and
3(19) 2 59
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11
How to Convert a Degree to Minute or Second?

• One minute is 1/60 of a degree.
• One second is 1/60 of a minute.

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Example

• Perform the calculation.
• Since 86 60 26, the sum is written

• Perform the calculation.
• Write

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Example

• Convert 36.624

• Convert

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How to Determine an Angle is in Standard
Position?

• An angle is in standard position if its vertex is
  at the origin and its initial side is along the
  positive x-axis.

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What are Quadrantal Angles?

• Angles in standard position having their terminal
  sides along the x-axis or y-axis, such as angles
  with measures 90, 180, 270, and so on, are
  called quadrantal angles.

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What are Coterminal Angles?

• A complete rotation of a ray results in an angle
  measuring 360. By continuing the rotation,
  angles of measure larger than 360 can be
  produced. Such angles are called coterminal
  angles.

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Example

• Find the angles of smallest possible positive
  measure coterminal with each angle.
• a) 1115 b) -187
• Add or subtract 360 as may times as needed to
  obtain an angle with measure greater than 0 but
  less than 360.
• a) b) -187 360 173

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What are Vertical Angles?

• Vertical Angles have equal measures.
• The pair of angles NMP and RMQ are vertical
  angles.

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Parallel Lines and Transversal

• Parallel lines are lines that lie in the same
  plane and do not intersect.
• When a line q intersects two parallel lines, q,
  is called a transversal.

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Important Angle Relationships
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Example of Finding Angle Measures

• Find the measure of each marked angle, given that
  lines m and n are parallel.
• The marked angles are alternate exterior angles,
  which are equal.

• One angle has measure
• 6x 4 6(21) 4 130
• and the other has measure 10x - 80 10(21) - 80
  130

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Angle Sum of a Triangle

• The sum of the measures of the angles of any
  triangle is 180.

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Example of Applying the Angle Sum

• The measures of two of the angles of a triangle
  are 52 and 65. Find the measure of the third
  angle, x.

• Solution
• The third angle of the triangle measures 63.

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Types of Triangles Angles

• Note The sum of the measures of the angles of
  any triangle is 180.

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Types of Triangles Sides

• Again, the sum of the measures of the angles of
  any triangle is 180.

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What are the Conditions for Similar Triangles?

• Corresponding angles must have the same measure.
• Corresponding sides must be proportional. (That
  is, their ratios must be equal.)

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Example of Finding Angle Measures

• Triangles ABC and DEF are similar. Find the
  measures of angles D and E.

• Since the triangles are similar, corresponding
  angles have the same measure.
• Angle D corresponds to angle A which 35
• Angle E corresponds to angle B which 33

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Example of Finding Side Lengths

• Triangles ABC and DEF are similar. Find the
  lengths of the unknown sides in triangle DEF.

• To find side DE.
• To find side FE.

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Example of Application

• The two triangles are similar, so corresponding
  sides are in proportion.
• The lighthouse is 48 m high.

• A lighthouse casts a shadow 64 m long. At the
  same time, the shadow cast by a mailbox 3 feet
  high is 4 m long. Find the height of the
  lighthouse.

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The Six Trigonometric Functions

• Let (x, y) be a point other the origin on the
  terminal side of an angle ? in standard position.
  The distance from the point to the origin is
• The six trigonometric functions of ? are defined
  as follows.

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Example of Finding Function Values

• The terminal side of angle ? in standard position
  passes through the point (12, 16). Find the
  values of the six trigonometric functions of
  angle ?.

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Example of Finding Function Values (cont.)

• Since x 12, y 16, and r 20, we have

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Another Example

• Find the six trigonometric function values of the
  angle ? in standard position, if the terminal
  side of ? is defined byx 2y 0, x 0.
• We can use any point on the terminal side of ?
  to find the trigonometric function values.

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Another Example (cont.)

• Choose x 2
• The point (2, -1) lies on the terminal side, and
  the corresponding value of r is

• Use the definitions

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Example of Finding Function Values with
Quadrantal Angles

• Find the values of the six trigonometric
  functions for an angle of 270.
• First, we select any point on the terminal side
  of a 270 angle. We choose (0, -1). Here x 0, y
  -1 and r 1.

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Undefined Function Values

• If the terminal side of a quadrantal angle lies
  along the y-axis, then the tangent and secant
  functions are undefined.
• If it lies along the x-axis, then the cotangent
  and cosecant functions are undefined.

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What are the Commonly Used Function Values?
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Reciprocal Identities
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Example of Finding Function ValuesUsing
Reciprocal Identities

• Find cos ? if sec ?
• Since cos ? is the reciprocal of sec ?

• Find sin ? if csc ?

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Signs of Function Values at Different Quadrants
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Identify the Quadrant

• Identify the quadrant (or quadrants) of any angle
  ? that satisfies tan ? gt 0 and cot ? gt 0.
• tan ? gt 0 in quadrants I and III
• cot ? gt 0 in quadrants I and III
• so, the answer is quadrants I and III

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Ranges of Trigonometric Functions

• For any angle ? for which the indicated functions
  exist
• 1. -1 sin ? 1 and -1 cos ? 1
• 2. tan ? and cot ? can equal any real number
• 3. sec ? -1 or sec ? 1 and
• csc ? -1 or csc ? 1.
• (Notice that sec ? and csc ? are never between
  -1 and 1.)

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Pythagorean Identities
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Quotient Identities
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Example of Other Function Values

• Find sin ? and cos ? if tan ? 4/3 and ? is in
  quadrant III.
• Since ? is in quadrant III, sin ? and cos ? will
  both be negative.
• sin ? and cos ? must be in the interval -1, 1.

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Example of Other Function Values (cont.)

• We use the identity

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What have we learned?

• We have learned to
• Use basic terms associated with angles.
• Find measures of complementary and supplementary
  angles.
• Calculate with degrees, minutes, and seconds.
• Convert between decimal degrees and degrees,
  minutes, and seconds.
• Identify the characteristics of an angle in
  standard position.
• Find measures of coterminal angles.
• Find angle measures by using geometric
  properties.
• Apply the angle sum of a triangle property.

http//faculty.valenciacc.edu/ashaw/ Click link
to download other modules.
Rev.S08
48
What have we learned? (Cont.)

1. Find angle measures and side lengths in similar
   triangles.
2. Solve applications involving similar triangles.
3. Learn basic concepts about trigonometric
   functions.
4. Find function values of an angle or quadrantal
   angles.
5. Decide whether a value is in the range of a
   trigonometric function
6. Use the reciprocal, Pythagorean and quotient
   identities.
7. Identify the quadrant of an angle.
8. Find other function values given one value and
   the quadrant.

http//faculty.valenciacc.edu/ashaw/ Click link
to download other modules.
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49
Credit

• Some of these slides have been adapted/modified
  in part/whole from the slides of the following
  textbook
• Margaret L. Lial, John Hornsby, David I.
  Schneider, Trigonometry, 8th Edition

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Rev.S08