Angles and their measure, Solving Right Triangle and Trigonometric Ratios

Basic Sciences Department

week 1

Outlines: 6.1: Angles and Their Measure
 6.2: Solving Right Triangles
 6.3: Trigonometric Functions: A Unit Circle
Approach

5 February 2014

PRECACULUS, Mathematics(2), MATH-112, McGraw Hill

Objectives

prepared by : Mais Obeidat

Angles and Their Measure

Section 6.1

prepared by : Mais Obeidat

6.1. Angles and Their Measure

 Angles
 Degree and Radian Measure

 Converting Degrees to Radians and vice
versa
Prepared by: Mais Obeidat


 Angles

The vertex



 Sign of Angle:

Counterclockwise

Clockwise

Positive angle

Negative angle



Two different angles that have the same initial and terminal sides, are
called coterminal.



If the terminal side of an angle in standard position along the coordinate
axis, the angle is said to be a quadrantal angle.


 Degree and Radian Measure
 Definition 1: Degree Measure

Example:


 Types of Angles:



A degree can divided using:

Converting from decimal degree (DD) to degree-minute-second form (DMS) and
vise versa:

Degree = 60 minute , minute = 60 second . Thus:


Example 1:

From DMS to DD and Back

Solutions:



Matched Problem 1:

Solutions:


 Definition 2: Radian Measure


Example 2:

Computing Radian Measure

Solutions:

= 4 radians.



Matched Problem 2:

Solutions:

= ………… radians.



 Converting Degree to Radians and Vice Versa


Example 3:

Radian - Degree Conversions

Solutions:

Exact
Exact
Four significant digits
Three significant digits


To two decimal places



Matched Problem 3:

Solutions:


Solving Right Triangles

Section 6.2

6.2. Solving Right Triangles

 Trigonometric Ratio.
 Evaluation of Trigonometric Ratio.
 Solving Right Triangles.

Prepared by : Mais Obeidat


 Trigonometric Ratios.
Satisfied that:

Why?
If only the angles of right triangle are known, it is impossible to solve the sides.
If we are given

One acute angle and a side

Two sides
Then

It is possible to solve the remaining three quantities. (This
process is called solving the right triangle)



Then, the triangles are similar and
ratios of corresponding sides are equal
Therefore

These six ratios, the trigonometric ratios, are called sine, cosine, tangent,
cosecant, secant, and cotangent.

 Trigonometric Ratios

SOHCAHTOA

 Right Triangle Ratios

Hypotenuse

Opposite

Adjacent

 Reciprocal Relationships

 Complementary Relationships

The trigonometric ratios, cosine, cosecant , and cotangent are sometimes referred
to as the cofunctions of sine , secant , and tangent, respectively.


Exact Values of the Trigonometric Functions( Standard angles)
 Evaluation of Trigonometric Ratio.
How?

How?
Why?


Example 1:

Calculator Evaluation

Solutions:
First, make certain that the calculator is set in degree mode .



Matched Problem 1:

Solutions:


 Solving Right Triangles.
Example 2:

Right Triangle Solution

Solutions:
First, draw a figure and label the parts.


The inverse of sine.

Matched Problem 1:

Solutions:

To the nearest hundredth degree
To the nearest minute

We use the same
process with the other 5
trigonometric functions

Example 3:

Right Triangle Solution

Solutions:
First, draw a figure and label the parts.


Matched Problem 3:

Solutions:


Trigonometric Functions: A Unit Circle
Approach
Section 6.3


6.3. Trigonometric Functions: A Unit Circle Approach

 The Wrapping Function.
 Definitions of the Trigonometric Functions.



 The Wrapping Function:

The point is called a circular point


1
1

1

(1,0)

(0,-1)

(0,1)

(1,0)

(-1,0)


Example 1:

Coordinates of Circular Points

Solutions:



Matched Problem 1:

Solutions:



 Coordinates of the key circular point

 Defining the Trigonometric Functions:
 Definition1: Trigonometric Functions

Remarks:

lies on the unit circle


Example 2:

Evaluating Trigonometric Functions

Solutions:



Matched Problem 2:


Example 3:

Solutions:

Calculator Evaluation

Sample questions( Test Your Self )
Choose the correct answer for the following questions:

Sample Questions

a) Quadrant III

b) Quadrant II

c) Quadrant I

c)0

d) Quadrant IV

d) Undefined


Home Work
• P. 8, #7, 14, 19,23,25,37,41,55.
• P.18 , # 10,18,21,29,35,34-42.
• P.30, #9,11,29,33,39, 47, 53-56,75,81.
Give all the same details as we did before .


Angles and their measure, Solving Right Triangle and Trigonometric Ratios

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