Chapter 4 Review of the Trigonometric Functions MAT 205 SPRING 2009 Chapter 4 Review of the Trigonometric Functions

Standard Position Vertex at origin The initial side of an angle in standard position is always located on the positive x-axis.

Positive and negative angles When sketching angles, always use an arrow to show direction.

Classifying Angles Angles in standard position are often classified according to the quadrant in which their terminal sides lie. Example: 50º is a 1st quadrant angle. 208º is a 3rd quadrant angle. II I -75º is a 4th quadrant angle. III IV

Classifying Angles Standard position angles that have their terminal side on one of the axes are called ______________ angles. For example, 0º, 90º, 180º, 270º, 360º, … are quadrantal angles.

Degrees, minutes, and seconds 1 minute (1') = degree OR 1° = ______ ' 1 second (1") = _____ minute OR 1' = _____" Therefore, 1 second = ________ degree Example Convert to decimal degrees (to three decimal places):

Degrees, minutes, and seconds Conversions between decimal degrees and degrees, minutes, seconds can be easily accomplished using your TI graphing calculator. The ANGLE menu on your calculator has built-in features for converting between decimal degrees and DMS. Note that the seconds () symbol is not in the ANGLE menu. Use  for  symbol.

Practice Using your TI graphing calculator, NOTE: SET MODE TO DEGREE Using your TI graphing calculator, 1) Convert to decimal degrees to the nearest hundredth of a degree. 2) Convert 57.328° to an equivalent angle expressed to the nearest second.

Angles  and  are coterminal. Coterminal Angles Angles that have the same initial and terminal sides are coterminal. Angles  and  are coterminal.

Examples of Coterminal Angles Find one positive and one negative coterminal angle for each angle given. a) 125 b) 240 34' c) -311.8

The sides of a right triangle MAT 205 SPRING 2009 The sides of a right triangle Take a look at the right triangle, with an acute angle, , in the figure below. Notice how the three sides are labeled in reference to .  Side adjacent to  Side opposite  Hypotenuse

Definitions of the Six Trigonometric Functions MAT 205 SPRING 2009 Definitions of the Six Trigonometric Functions To remember the definitions of Sine, Cosine and Tangent, we use the acronym : “SOH CAH TOA”

Definitions of Trigonometric Functions of an Angle MAT 205 SPRING 2009 Definitions of the Trig Functions Definitions of Trigonometric Functions of an Angle Let  be an angle in standard position with (x, y) a point on the terminal side of  and r is the distance from the origin to the point. Using the Pythagorean theorem, we have .  y x (x, y) r

MAT 205 SPRING 2009 Example Let (12, 5) be a point on the terminal side of . Find the value of the six trig functions of . y r 5 x  (12, 5) 12 First you must find the value of r:

MAT 205 SPRING 2009 Example (cont) r 5 x  (12, 5) 12

MAT 205 SPRING 2009 Example Given that  is an acute angle and , find the exact value of the five remaining trig functions of .

MAT 205 SPRING 2009 Example Find the value of tan  given csc  = 1.02, where  is an acute angle. Give answer to three significant digits.

Special Right Triangles MAT 205 SPRING 2009 Special Right Triangles The 45º- 45º- 90º Triangle Ratio of the sides: Find the exact values of the six trig functions for 45 sin 45 = csc 45 = cos 45 = sec 45 = tan 45 = cot 45 = 1 45º

Special Right Triangles MAT 205 SPRING 2009 Special Right Triangles The 30º- 60º- 90º Triangle Ratio of the sides: Find the exact values of the six trig functions for 30 sin 30 = csc 30 = cos 30 = sec 30 = tan 30 = cot 30 = 1 60º 30º 2

Special Right Triangles MAT 205 SPRING 2009 Special Right Triangles The 30º- 60º- 90º Triangle Ratio of the sides: Find the exact values of the six trig functions for 60 sin 60 = csc 60 = cos 60= sec 60 = tan 60 = cot 60 = 1 60º 30º 2

Using the calculator to evaluate trig functions MAT 205 SPRING 2009 Using the calculator to evaluate trig functions Make sure the MODE is set to the correct unit of angle measure (i.e. Degree vs. Radian) Example: Find to three significant digits.

Using the calculator to evaluate trig functions MAT 205 SPRING 2009 Using the calculator to evaluate trig functions For reciprocal functions, you may use the  button, but DO NOT USE THE INVERSE FUNCTIONS (e.g. )! Example: 1. Find 2. Find (to 3 significant dig) (to 4 significant dig)

Angles and Accuracy of Trigonometric Functions MAT 205 SPRING 2009 Angles and Accuracy of Trigonometric Functions Measurement of Angle to Nearest Accuracy of Trig Function 1° 2 significant digits 0. 1° or 10’ 3 significant digits 0. 01° or 1’ 4 significant digits

MAT 205 SPRING 2009 The inverse trig functions give the measure of the angle if we know the value of the function. Notation: The inverse sine function is denoted as sin-1x or arcsinx. It means “the angle whose sine is x”. The inverse cosine function is denoted as cos-1x or arccosx. It means “the angle whose cosine is x”. The inverse tangent function is denoted as tan-1x or arctanx. It means “the angle whose tangent is x”.

is the angle whose sine is MAT 205 SPRING 2009 Example is the angle whose sine is Think of the related statement  must be 30°, therefore

MAT 205 SPRING 2009 Examples Evaluate the following inverse trig functions using the calculator. Give answer in degrees. To nearest 0.1 To 2 sig. dig.

MAT 205 SPRING 2009 Examples Evaluate the following inverse trig functions using the calculator. Give answer in degrees.

Example Solve the right triangle with the indicated measures. Solution MAT 205 SPRING 2009 Example Solve the right triangle with the indicated measures. A= C B b c a= Solution Answers:

MAT 205 SPRING 2009 Example A C B b c= a= Solution

Angle of Elevation and Angle of Depression MAT 205 SPRING 2009 Angle of Elevation and Angle of Depression The angle of elevation for a point above a horizontal line is the angle formed by the horizontal line and the line of sight of the observer at that point. The angle of depression for a point below a horizontal line is the angle formed by the horizontal line and the line of sight of the observer at that point. Horizontal line Angle of depression Angle of elevation Horizontal line