College Trigonometry 2 Credit hours through KCKCC or Donnelly

1
Chapter 2Acute Angles and Right Triangles
Section 2.1 Acute Angles
Section 2.2 Non-Acute Angles
Section 2.3 Using a Calculator
Section 2.4 Solving Right Triangles
Section 2.5 Further Applications
2
Section 2.1 Acute Angles

• In this section we will
• Define right-triangle-based trig functions
• Learn co-function identities
• Learn trig values of special angles

3
Right-Triangle-Based Definitions

• sin A csc A
• cos A sec A
• tan A cot A

y r
opp hyp
hyp opp
r y
x r
hyp adj
r x
adj hyp
opp adj
y x
adj opp
x y
4
Co-function Identities

• sin A cos(90à- A) csc A sec(90à- A)
• cos A sin(90à- A) sec A csc(90à- A)
• tan A cot(90à- A) cot A tan(90à- A)

5
Special Trig Values
0à 30à 45à 60à 90à
sin ñ0 2 ñ1 2 ñ2 2 ñ3 2 ñ4 2
cos ñ4 2 ñ3 2 ñ2 2 ñ1 2 ñ0 2
tan 0 ñ3 3 1 ñ3 Und
csc 2 ñ0 2 ñ1 2 ñ2 2 ñ3 2 ñ4
sec 2 ñ4 2 ñ3 2 ñ2 2 ñ1 2 ñ0
cot Und ñ3 1 ñ3 3 0
6
Special Trig Values
0à 30à 45à 60à 90à
sin 0 1 2 ñ2 2 ñ3 2 1
cos 1 ñ3 2 ñ2 2 1 2 0
tan 0 ñ3 3 1 ñ3 Und
csc Und 2 ñ2 2ñ3 3 1
sec 1 2ñ3 3 ñ2 2 Und
cot Und ñ3 1 ñ3 3 0
7
Section 2.2 Non-Acute Angles

• In this section we will learn
• Reference angles
• To find the value of any non-quadrantal angle

8
Reference Angles

• in Quad I
• in Quad II
• in Quad III
• in Quad IV

Quadrant I (,)
Quadrant II (-,)
Quadrant III (-,-)
Quadrant IV (,-)
9
Reference Angle for in (0à,360à)

• 0à
• 180à -
• 180à
• 360à -

Quadrant I (,)
Quadrant II (-,)
Quadrant III (-,-)
Quadrant IV (,-)
10
Finding Values of AnyNon-Quadrantal Angle

1. If gt 360à, or if lt 0à, find a coterminal
   angle by adding or subtracting 360à as many times
   as needed to get an angle between 0à and 360à.
2. Find the reference angle .
3. Find the necessary values of the trigonometric
   functions for the reference angle .
4. Determine the correct signs for the values found
   in Step 3 thus giving you .

11
Section 2.3 Using a Calculator

• In this section we will
• Approximate function values using a calculator
• Find angle measures using a calculator

http//mathbits.com/mathbits/TISection/Openpage.ht
m
12
Approximating function values

• Convert 57º 45' 17'' to decimal degrees
• In either Radian or Degree Mode    Type 57º 45'
  17'' and hit Enter.        º is under Angle
  (above APPS) 1        '  is under Angle (above
  APPS) 2        ''  use ALPHA (green) key with
  the quote symbol above the sign.               
                                                    
                                        Answer 
  57.75472222

13
Approximating function values

• Convert 48.555º to degrees, minutes, seconds
• Type 48.555 ?DMS                         
  Answer  48º 33' 18''        The ?DMS is 4 on
  the Angle menu (2nd APPS).  This function works
  even if Mode is set to Radian.

14
Finding Angle Measures

• Given cos A .0258.  Find / A expressed in
  degree, minutes, seconds.
• With the mode set to Degree  
• Type cos-1(.0258). 
• Hit Enter.
• Engage ?DMS     Answer  88º 31' 17.777''   (Be
  careful here to be in the correct mode!!)

15
Section 2.4 Solving Right Triangles

• In this section we will
• Understand the use of significant digits in
  calculations
• Solve triangles
• Solve problems using angles of Elevation and
  Depression

16
Significant Digits In Calculations

• A significant digit is a digit obtained by actual
  measurement.
• An exact number is a number that represents the
  result of counting, or a number that results from
  theoretical work and is not the result of a
  measurement.

17
Significant Digits for Angles
Number of Significant Digits Angle Measure to the Nearest
2 Degree
3 Ten minutes, or nearest tenth of a degree
4 Minute, or nearest hundredth of a degree
5 Tenth of a minute, or nearest thousandth of a degree
18
Solving Triangles

• To solve a triangle find all of the remaining
  measurements for the missing angles and sides.
• Use common sense. You dont have to use trig for
  every part. It is okay to subtract angle
  measurements from 180à to find a missing angle or
  use the Pythagorean Theorem to find a missing
  side.

19
Looking Ahead

• The derivatives of parametric equations, like x
  f(t) and y g(t) , often represent rate of
  change of physical quantities like velocity.
  These derivatives are called related rates since
  a change in one causes a related change in the
  other.
• Determining these rates in calculus often
  requires solving a right triangle.

20
Angle of Elevation

Angle of elevation

Horizontal eye level
21
Angle of depression

Horizontal eye level

Angle of depression
22
Section 2.5 Further Applications

• In this section we will
• Discuss Bearing
• Work with further applications of solving
  non-right triangles

23
Bearings

• Bearings involve right triangles and are used to
  navigate. There are two main methods of
  expressing bearings
• Single angle bearings are always measured in a
  clockwise direction from due north
• North-south bearings always start with N or S and
  are measured off of a North-south line with acute
  angles going east or west so many degrees so they
  end with E or W.

24
First Method

45à
330à
135à
25
Second Method

N



S
N 30à W
N 45à E
S 45à E