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 2 Acute 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 = = opp hyp adj hyp opp adj hyp opp hyp adj opp ryry rxrx xyxy yryr xrxr yxyx

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 0à0à30à45à60à90à sin ñ0 2ñ1 2ñ2 2ñ3 2ñ4 2 cos ñ4 2ñ3 2 ñ2 2 ñ1 2ñ0 2 tan0 ñ3 3 1ñ3 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 ñ31 3 0

Special Trig Values 0à0à30à45à60à90à sin ñ2 2ñ3 2 1 cos1 ñ3 2 ñ tan0 ñ3 3 1ñ3 Und csc Und 2ñ2 2ñ3 3 1 sec1 2ñ3 3 ñ22 Und cot Und ñ31 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 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 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

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:

Approximating function values Convert º to degrees, minutes, seconds: Type ►DMS Answer: 48º 33' 18'' 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 = 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' '' (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: 2Degree 3Ten minutes, or nearest tenth of a degree 4Minute, or nearest hundredth of a degree 5Tenth 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 £ 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 330à 135à

Second Method N S N £ £ £ N 45à E N 30à W S 45à E