LESSON 2 DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS ...

[Pages:18]LESSON 2 DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING THE UNIT CIRCLE

Topics in this lesson:

1. DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING A

CIRCLE OF RADIUS r

2. DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING THE

UNIT CIRCLE

3. THE SPECIAL ANGLES IN TRIGONOMETRY

4. TEN THINGS EASILY OBTAINED FROM UNIT CIRCLE

TRIGONOMETRY

5. THE SIX TRIGONOMETRIC FUNCTIONS OF THE THREE SPECIAL

ANGLES IN THE FIRST QUADRANT BY ROTATING

COUNTERCLOCKWISE

6. ONE METHOD TO REMEMBER THE TANGENT OF THE SPECIAL

ANGLES OF

6

(30) ,

4

( 45) ,

AND

3

( 60 )

7. THE SIX TRIGONOMETRIC FUNCTIONS OF THE REST OF THE

SPECIAL ANGLES

1. DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING A CIRCLE OF RADIUS r

y

Pr ( ) ( x , y )

r s

- r

r

x

r

- r

x2 y2 r2

Copyrighted by James D. Anderson, The University of Toledo math.utoledo.edu/~janders/1330

Definition Let Pr ( ) ( x , y ) be the point of intersection of the terminal side of the angle with the circle whose equation is x 2 y 2 r 2 . Then we define the

following six trigonometric functions of the angle

cos x r

sin y r

sec

r x

,

provided that

x

0

csc

r y,

provided that

y

0

tan

y x,

provided that

x

0

cot

x y,

provided that

y

0

NOTE: By definition, the secant function is the reciprocal of the cosine function. The cosecant function is the reciprocal of the sine function. The cotangent function is the reciprocal of the tangent function.

Back to Topics List

2. DEFINITION OF THE SIX TRIGONOMETRIC FUNCTIONS USING THE UNIT CIRCLE

Since you can use any size circle to define the six trigonometric functions, the best circle to use would be the Unit Circle, whose radius r is 1. Using the Unit Circle, we get the following special definition.

Definition Let P ( ) ( x , y ) be the point of intersection of the terminal side of

the angle with the Unit Circle. Since r 1 for the Unit Circle, then by the definition above, we get the following definition for the six trigonometric functions of the angle using the Unit Circle

cos x

sec

1 x

,

provided that x

0

Copyrighted by James D. Anderson, The University of Toledo math.utoledo.edu/~janders/1330

sin y

tan

y x,

provided that

x

0

csc

1 y,

provided that

y

0

cot

x y,

provided that

y

0

P ( ) ( x, y )

- 1

y 1

1 - 1

s 1 x

x2 y 2 1 (The Unit Circle)

NOTE: The definition of the six trigonometric functions of the angle in terms of the Unit Circle says that the cosine of the angle is the x-coordinate of the point of intersection of the terminal side of the angle with the Unit Circle. This definition also says that the sine of the angle is the y-coordinate of the point of intersection of the terminal side of the angle with the Unit Circle. The tangent of the angle is the y-coordinate of the point of intersection of the terminal side of the angle with the Unit Circle divided by the x-coordinate of the point of intersection. The secant function is still the reciprocal of the cosine function, the cosecant function is still the reciprocal of the sine function, and the cotangent function is still the reciprocal of the tangent function.

Examples Find the exact value of the six trigonometric functions for the following angles.

1. 0

Copyrighted by James D. Anderson, The University of Toledo math.utoledo.edu/~janders/1330

P ( 0) (1, 0)

cos 0 x 1 sin 0 y 0 tan 0 y 0 0

x1

sec 0 1 1 1 cos 0 1

csc 0 1 sin 0

1 0

= undefined

cot 0

1 tan 0

1 0

=

undefined

2.

2

NOTE: This is the 9 0 angle in units of degrees.

P (0, 1) 2

Copyrighted by James D. Anderson, The University of Toledo math.utoledo.edu/~janders/1330

cos x 0 2

sec 2

1 cos

1 0

undefined

2

sin y 1 2

csc 2

1 sin

1 1

1

2

tan

2

y x

1 0

=

undefined

cot 1 x 0 0 2 tan y 1 2

3.

NOTE: This is the 18 0 angle in units of degrees.

P ( ) ( 1, 0)

cos x 1

sec 1 1 1 cos 1

sin y 0

csc

1 sin

1 0

=

undefined

Copyrighted by James D. Anderson, The University of Toledo math.utoledo.edu/~janders/1330

tan y 0 0 x 1

cot

1 tan

1 0

= undefined

4. 270 3

NOTE: This is the 2 angle in units of radians.

P ( 270 ) ( 0, 1)

cos 270 0

sin 270 1

tan 270

1 0

=

undefined

sec 270

1 0

=

undefined

csc 270 1

cot 270 0 0 1

5. 2

NOTE: This is the 36 0 angle in units of degrees.

Copyrighted by James D. Anderson, The University of Toledo math.utoledo.edu/~janders/1330

P ( 2 ) (1, 0)

cos 2 1 sin 2 0 tan 2 0 0

1

sec 2 1

csc 2

1 0

=

undefined

cot 2

1 0

=

undefined

6. 90

NOTE:

This is the

2

angle in units of radians.

P ( 90 ) ( 0, 1)

Copyrighted by James D. Anderson, The University of Toledo math.utoledo.edu/~janders/1330

cos ( 90 ) 0

sin ( 90 ) 1

tan ( 90 )

1 0

=

undefined

sec ( 90 )

1 0

=

undefined

csc ( 90 ) 1

cot ( 90 ) 0 0 1

7. 180 NOTE: This is the angle in units of radians.

P ( 180 ) ( 1, 0)

cos ( 180 ) 1

sec ( 180 ) 1

sin ( 180 ) 0

csc ( 180 )

1 0

=

undefined

tan ( 180 ) 0 0 1

cot ( 180 ) 1 0

=

undefined

Copyrighted by James D. Anderson, The University of Toledo math.utoledo.edu/~janders/1330

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download