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

Chapter 2Acute Angles and Right Triangles

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

sin A = = csc A = =

cos A = = sec A = =

tan A = = cot A = =

opphyp

adjhyp

oppadj

hypopp

hypadj

adjopp

ry

rx

xy

yr

xr

yx

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 Values0à 30à 45

à60à 90à

sin ñ0 2 ñ1 2 ñ2 2 ñ3 2 ñ4 2

cos ñ4 2 ñ32

ñ22

ñ1 2 ñ0 2

tan 0 ñ33

1 ñ3 Und

csc 2ñ0

2ñ1

2ñ2

2ñ3

2ñ4

sec 2ñ4

2ñ3

2ñ2

2ñ1

2ñ0

cot Und ñ3 1 ñ33

0

Special Trig Values0à 30à 45

à60à 90à

sin 0 12

ñ2 2 ñ3 2 1

cos 1 ñ32

ñ22

12

0

tan 0 ñ33

1 ñ3 Und

csc Und 2 ñ2 2ñ33

1

sec 1 2ñ33

ñ2 2 Und

cot Und ñ3 1 ñ33

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 I

(+,+)

Quadrant II

(-,+)

Quadrant III

(-,-)

Quadrant IV

(+,-)

£’

£’£’

£’

Reference Angle £’ for £ in (0à,360à)

£’= 0à + ££’= 180à - ££’= 180à + ££’= 360à - £

Quadrant I

(+,+)

Quadrant II

(-,+)

Quadrant III

(-,-)

Quadrant IV

(+,-)

£

££

££’

£’ £’

Finding Values of AnyNon-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

£

Angle of elevation

Horizontal eye level

Angle of depression

£ 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 330à135à

Second Method

N

S

N

£ £ £

N 45à E N 30à W

S 45à E