Chapter 2
Acute 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
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
Right-Triangle-Based
Definitions
y
opp
= csc A = = r
hyp
hyp
opp
r
y
cos A = =
x
adj
sec A = = r
hyp
hyp
adj
r
x
tan A = =
opp
cot A = = y
adj
x
adj
opp
x
y
sin A =
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)
Special Trig Values
0à 30à 45à 60à 90à
sin
ñ0 2
ñ1 2
ñ2 2
ñ3 2
ñ4 2
cos
ñ4 2
ñ2
2
ñ1 2
ñ0 2
tan
0
1
ñ3
Und
csc
2
ñ0
2
ñ4
ñ3
2
ñ3
3
2
ñ1
2
ñ3
2
ñ2
2
ñ2
2
ñ3
2
ñ1
2
ñ4
2
ñ0
Und
ñ3
1
ñ3
3
0
sec
cot
Special Trig Values
sin
0à 30à 45à 60à 90à
1
ñ2 2 ñ3 2
0
1
cos
1
tan
0
csc
Und
sec
cot
2
ñ3
2
ñ3
3
ñ2
2
1
2
0
1
ñ3
Und
2
ñ2
2ñ3
3
1
1
2ñ3
3
ñ2
2
Und
Und
ñ3
1
ñ3
3
0
Section 2.2 Non-Acute Angles
In this section we will learn:
• Reference angles
• To find the value of any non-quadrantal
angle
Reference Angles
£ in Quad I
£ in Quad II
£ in Quad III
£ in Quad IV
Quadrant II
Quadrant I
(-,+)
(+,+)
£’
£’
£’
£’
Quadrant III
Quadrant IV
(-,-)
(+,-)
Reference Angle £’ for £ in (0à,360à)
Quadrant I
(-,+)
(+,+)
£’
£’
£
£
£
£’
£’= 0à + £
£’= 180à - £
£’= 180à + £
£’= 360à - £
Quadrant II
£
Quadrant III
Quadrant IV
(-,-)
(+,-)
Finding Values of Any
Non-Quadrantal Angle
1. If £ > 360à, or if £ < 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 £.
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.htm
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
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.
Finding Angle Measures
Given cos A = .0258. Find / A expressed
in degree, minutes, seconds.
• With the mode set to Degree:
1. Type cos-1(.0258).
2. Hit Enter.
3. Engage ►DMS Answer: 88º 31' 17.777''
(Be careful here to be in the correct mode!!)
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
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.
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
Solving Triangles
• To solve a triangle find all of the remaining
measurements for the missing angles and
sides.
• Use common sense. You don’t 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.
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.
Angle of Elevation
£
Horizontal eye level
Angle of depression
Horizontal eye level
£
Section 2.5 Further Applications
In this section we will:
• Discuss Bearing
• Work with further applications of solving
non-right triangles
Bearings
• Bearings involve right triangles and are
used to navigate. There are two main
methods of expressing bearings:
1. Single angle bearings are always measured
in a clockwise direction from due north
2. 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.
First Method
N
N
N
£
£
45à
135à
£
330à
Second Method
N
N
£
£
£
S
N 45à E
S 45à E
N 30à W