Trigonometric Functions

The Unit Circle

The Unit Circle Definition: A circle whose center is the origin and whose radius

has a length of one.

Based on the definition, give the coordinates for the x- and y-intercepts for the diagram below.

(__, __)

(__, __)

(__, __)

(__, __)

1 , 0

0 , 1

-1 , 0

0 , -1

The Unit Circle Let’s determine the coordinates on the unit circle for a 45 angle.

2h l

45

(__, __)

Drop the perpendicular from the pointon the circle to the positive x-axis.

What type of triangle is created?An isosceles right triangle

What is the length of the hypotenuse? Why?

1 because it is a unit circle

1

What is the relationship between thehypotenuse and a leg of an isosceles right triangle?

What is the length of each leg?1 2

22l

What is the x-coordinate of the point? 2

22

2The y-coordinate?

2

2

2

2

2

2

2

2

The Unit Circle

2

2

2

2 2

2

135

(__, __)

What would be the coordinates of the point on the circle if we draw a 135 angle? Explain.

What would be the coordinates of the point on the circle if we draw a 225 angle? Explain.

2

2(__, __)

225

The Unit Circle

2

2

What would be the coordinates of the point on the circle if we draw a 315 angle? Explain.

315

(__, __)2

2

The same procedure can be used to findthe coordinates for a 30 angle and a 60 angle. Visiting all quadrants would

result in the following figure.

The Trigonometric Functions Let t be the measure of a central angle and let (x, y) be the point on the unit

circle corresponding to t. The following are the definitions for the six trigonometric functions based on the unit circle.

sin t y 1csc t

y

cos t x 1sec t

x

tany

tx

cotx

ty

The Trigonometric Functions

1sin

2t y

6t

3 1,

2 2

Step 1: Identify the quadrant the terminal side of the angle

is located. Step 2: Identify the coordinates that correspond with that angle.

How to determine the trigonometric values for a given angle

Ex. 1: Evaluate the six trigonometric functions for

Quadrant I

Step 3: Follow the definitions

3cos

2t x 1

csc 2ty

1 2 2 3sec

33tx

32

12

cot 3x

ty

12

32

1 3tan

33

ytx

The Trigonometric Functions

5

4t

2 2,

2 2

2sin

2t y

Ex. 2: Evaluate the six trigonometric functions for

Step 1: Identify the quadrant the terminal side of the angle is located.

Quadrant II

Step 2: Identify the coordinates that correspond with that angle.

Step 3: Follow the definitions

1 2csc 2

2ty

2cos

2t x

22

22

tan 1y

tx

1 2sec 2

2tx

22

22

cot 1x

ty

The Trigonometric Functions

7

3t

7 7 62

3 3 3

1 3,

2 2

Ex. 2: Evaluate the six trigonometric functions for

In this problem the value of the angle exceeds 2. Find the coterminal anglewhose value lies between 0 and 2

Step 1: Identify the quadrant the terminal side of the angle is located.

Quadrant I

Step 2: Identify the coordinates that correspond with that angle.

Step 3: Follow the definitions

1cos

2t x

3sin

2t y tan 3

ytx

1 2 3csc

3ty

1sec 2t

x

3cot

3

xty

The Trigonometric Functions Recapping for evaluating the six trigonometric

functions for any given angle. If the given angle does not lie between 0 and 2, find its

simplest positve coterminal angle and use that value. Identify the quadrant the terminal side for the given angle lies. Determine the coordinates on the unit circle for the given

angle. Follow the definitions for the six trigonometric functions.

Even and Odd Trig Functions What is an even function?

An even function is when one substitutes a negative value into the original function and the outcome is the same as its positive value.

f (- x) = f (x)

Ex. f (x) = x2

f (2) = 22 = 4

f (-2) = (-2)2 =4

Therefore, f (-2) = f (2) and f (x) = x2 is an even function.

Even and Odd Trig Functions What is an odd function?

An odd function is when one substitutes a negative value into the original function and the outcome is the opposite of its positive value.

f (- x) = - f (x)

Ex. f (x) = x3

f (2) = 23 = 8

f (-2) = (-2)3 = - 8

Therefore, f (-2) = - f (2) and f (x) = x3 is an odd function.

Even and Odd Functions Let’s determine whether or not the sine function is even or odd.

f(x) = sin x

What is the sin 30? What is the sin (- 30)? sin (- 30) = - sin (30) sin (- 30) = -( ½ ) = - ½

Because the sin (- 30) = - sin (30), the sine function is odd.

sin (-x) = - sin x

½ - ½

Even and Odd Trig Functions Lets determine whether the cosine function is even or odd.

f (x) = cos x

What is the cos 60? What is the cos (- 60)? cos (- 60) = cos 60 cos (- 60) = ½

Because the cos (- 60) equals cos 60, cosine is an even function

cos (- x) = cos x

½½

Even and Odd Trig Functions Determine what the other 4 trig functions are – even or odd

tan (x)

csc (x)

sec (x)

cot (x)

odd function

odd function

odd function

even function

Evaluating Trig Fuctions with a Calculator

Steps Set the calculator into the correct mode

Casio From the MENU select RUN Press SHIFT, then MENU (above MENU you should see SET

UP) Scroll down and highlight Angle Select F1 for Deg, Select F2 for Rad Press EXE (Blue key at bottom of calculator) Type the measure of the angle into the calculator Press EXE

Evaluating Trig Fuctions with a Calculator

Using the calculator find the values for the following (round to 4 decimal places): 1. sin 214 2. tan (-175 ) 3. cos 5/9 4. sin 14/5

-0.5592

0.0875

-0.1736

0.5878

Evaluating Trig Fuctions with a Calculator

To evaluate cosecant, secant, or cotangent on the calculator, follow these steps:

Casio Make sure you are in the right mode (degree or radian) In parentheses type the trig function on the calculator that is the

reciprocal function of the one being evaluated. On the outside of the parentheses, press SHIFT, then ). A - 1 should

appear next to the parentheses. Example: Evaluate csc 40

Make sure you are in the degree mode. Type into the calculator (sin 40) Press SHIFT, then ). Your screen should now read (sin 40) - 1

Press EXE. Your answer should be 1.5557

Evaluating Trig Fuctions with a Calculator

Using the calculator find the values for the following (round to 4 decimal places): 1. sec 297 2. cot 19/12 3. csc (- 11.78) 4. sec /2

2.2027

-0.2679

1.4128

Ma ERROR - WHY?

4.2 Trigonometric Fuctions:

The Unit Circle Can you

Sketch the unit circle and place key angles and coordinates on it?

Explain how these coordinates are derived using either 30, 45 , or 60 ?

Determine the trig value for certain angles using the unit circle?

Explain why a trig function is either even or odd? Evaluate a trig function using the calculator?