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 sin A = = csc A = = cos A = = sec A = = hyp x r tan A = = cot A = = x hyp opp hyp adj adj opp x y r x r y

Co-function Identities sin A = cos(90 à- A) c sc A = sec(90 à- A) cos A = sin(90 à- A) sec A = csc(90 à- A) tan A = cot(90 à- A) cot A = tan(90 à- A)

sin Special Trig Values

0 à 30à 45à 60à 90à

ñ0 2 ñ1 2 ñ2 2 ñ3 2 ñ4 2 cos ñ4 2 tan 0 csc sec cot 2 ñ0 2 ñ4 Und ñ3 2 ñ3 3 2 ñ1 2 ñ3 ñ3 ñ2 2 1 2 ñ2 2 ñ2 1 ñ1 2 ñ0 2 ñ3 2 ñ3 2 ñ1 ñ3 3 Und 2 ñ4 2 ñ0 0

sin cos tan Special Trig Values

0 à 30à 45à 60à 90à

0 1 0 1 2 ñ3 2 ñ3 3 ñ2 2 ñ3 2 ñ2 2 1 1 2 ñ3 1 0 Und csc Und 2 ñ2 2 ñ3 3 1 sec cot 1 Und 2 ñ3 3 ñ3 ñ2 1 2 ñ3 3 Und 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 Quadrant II (-,+) £ in Quad I £ in Quad II £ in Quad III £ in Quad IV £ ’ Quadrant III (-,-) £ ’ Quadrant I (+,+) £ ’ £ ’ Quadrant IV (+,-)

Reference Angle £ ’ for £ in (0à,360à) £ ’ = 0 à + £ £ ’ = 180

à

£ ’ = 180

à +

£ ’ = 360

à

£ £ £ Quadrant II (-,+) £ £ ’ ’ £ £ Quadrant III (-,-) Quadrant I (+,+) £ ’ £ £ Quadrant IV (+,-)

Finding Values of Any Non-Quadrantal Angle 1. If £ > 360 à, or if £ < 0 à, find a coterminal angle by adding or subtracting many times as needed to get an angle between 0 à and 360 à.

360 à as 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 18'' Answer: 48º 33' The ►DMS is #4 on the Angle menu (2 nd 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

2 3 4 5

Angle Measure to the Nearest:

Degree Ten minutes, or nearest tenth of a degree Minute, or nearest hundredth of a degree 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 180 à 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.

N

£

45

à First Method

N N

£

135

à £

330 à

N

£

N 45

à

E

Second Method £ £

N S S 45

à

E N 30

à

W