rev.s08 mac 1114 module 1 trigonometric functions

Rev.S08

MAC 1114

Module 1

Trigonometric Functions

2Rev.S08

Learning Objectives

Upon completing this module, you should be able to:

1. Use basic terms associated with angles.

2. Find measures of complementary and supplementary angles.

3. Calculate with degrees, minutes, and seconds.

4. Convert between decimal degrees and degrees, minutes, and seconds.

5. Identify the characteristics of an angle in standard position.

6. Find measures of coterminal angles.

7. Find angle measures by using geometric properties.

8. Apply the angle sum of a triangle property.

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

http://faculty.valenciacc.edu/ashaw/

3Rev.S08

Learning Objectives (Cont.)

9. Find angle measures and side lengths in similar triangles.

10. Solve applications involving similar triangles.

11. Learn basic concepts about trigonometric functions.

12. Find function values of an angle or quadrantal angles.

13. Decide whether a value is in the range of a trigonometric function

14. Use the reciprocal, Pythagorean and quotient identities.

15. Identify the quadrant of an angle.

16. Find other function values given one value and the quadrant.



4Rev.S08

Trigonometric Functions


- Angles- Angles

- Angle Relationships and Similar Triangles- Angle Relationships and Similar Triangles

- Trigonometric Functions- Trigonometric Functions

- Using the Definitions of the Trigonometric - Using the Definitions of the Trigonometric FunctionsFunctions

There are four major topics in this module:


5Rev.S08

What are the basic terms?


Two distinct points determine a line called line AB.

Line segment AB—a portion of the line between A and B, including points A and B.

Ray AB—portion of line AB that starts at A and continues through B, and on past B.

A B

A B

A 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?


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

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


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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.

k − 16

k +20

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.

6x + 7 3x + 2

These angle measures are 6(19) + 7 = 121 and 3(19) + 2 = 59


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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 Convert 36.624


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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 1115 3(360 )−o

35 oo


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What are Vertical Angles?


Vertical Angles have equal measures.

The pair of angles NMP and RMQ are vertical angles.

M

QR

PN


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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.

m

n

parallel lines

qTransversal


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Important Angle Relationships


Angle measures are equal.2 & 6, 1 & 5, 3 & 7, 4 & 8

Corresponding angles

Angle measures add to 180.4 and 6

3 and 5

Interior angles on the same side of the transversal

Angle measures are equal.1 and 8

2 and 7

Alternate exterior angles

Angles measures are equal.4 and 5

3 and 6

Alternate interior angles

RuleAnglesName

m

n

q


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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.

m

n(10x − 80)

(6x + 4)

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.

52

65

x


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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

A

C B

F E

D

35

112 33

112


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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.

A

C B

F E

D

35

112 33

112

32

48

64

16

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.

64

4

3

x


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The Six Trigonometric Functions


Let (x, y) be a point other the origin on the terminal side of an angle θ in standard position. The distance from the point to the origin is

The six trigonometric functions of θ are defined as follows.


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Example of Finding Function Values


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

(12, 16)

16

12

θ


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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 θ in standard position, if the terminal side of θ is defined byx + 2y = 0, x ≥ 0.

We can use any point on the terminal side of θ 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?


undefined1undefined010360

−1undefined0undefined0−1270

undefined−1undefined0−10180

1undefined0undefined0190

undefined1undefined0100

csc θsec θcot θtan θcos θsin θθ

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Reciprocal Identities


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Example of Finding Function ValuesUsing Reciprocal Identities


Find cos θ if sec θ =

Since cos θ is the reciprocal of sec θ

Find sin θ if csc θ

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Signs of Function Values at Different Quadrants


−+−−+−IV

−−++−−III

+−−−−+II

++++++I

csc θsec θcot θtan θcos θsin θθ in Quadrant

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Identify the Quadrant


Identify the quadrant (or quadrants) of any angle θ that satisfies tan θ > 0 and cot θ > 0.

tan θ > 0 in quadrants I and III cot θ > 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 θ for which the indicated functions exist:

1. −1 ≤ sin θ ≤ 1 and −1 ≤ cos θ ≤ 1; 2. tan θ and cot θ can equal any real number; 3. sec θ ≤ −1 or sec θ ≥ 1 and

csc θ ≤ −1 or csc θ ≥ 1.

(Notice that sec θ and csc θ 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 θ and cos θ if tan θ = 4/3 and θ is in quadrant III.

Since θ is in quadrant III, sin θ and cos θ will both be negative.

sin θ and cos θ 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

1. Use basic terms associated with angles.

2. Find measures of complementary and supplementary angles.

3. Calculate with degrees, minutes, and seconds.

4. Convert between decimal degrees and degrees, minutes, and seconds.

5. Identify the characteristics of an angle in standard position.

6. Find measures of coterminal angles.

7. Find angle measures by using geometric properties.

8. Apply the angle sum of a triangle property.


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What have we learned? (Cont.)

9. Find angle measures and side lengths in similar triangles.

10. Solve applications involving similar triangles.

11. Learn basic concepts about trigonometric functions.

12. Find function values of an angle or quadrantal angles.

13. Decide whether a value is in the range of a trigonometric function

14. Use the reciprocal, Pythagorean and quotient identities.

15. Identify the quadrant of an angle.

16. Find other function values given one value and the quadrant.


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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


rev.s08 mac 1114 module 1 trigonometric functions

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