5.2 - Trigonometric Function of Real Numbers

Section 5.2 Trigonometric

Function of Real Numbers

Chapter 5 – Trigonometric Functions: Unit Circle Approach

5.2 - Trigonometric Function of Real Numbers

Definition of the Trig Functions

5.2 - Trigonometric Function of Real Numbers

Example – pg. 384Find sin t and cos t for the values of t whose terminal

points are shown on the unit circle in the figure. t increases in increments of /4.

5.2 - Trigonometric Function of Real Numbers

Special Values

5.2 - Trigonometric Function of Real Numbers

Domains of the Trig Functions

Function Domainsin, cos All real numbers

tan, sec All real numbers other than for any integer n

cot, csc All real numbers other than n for any integer n

2n

5.2 - Trigonometric Function of Real Numbers

Signs of the Trig Functions

Quad Positive Functions Negative Functions I All None II sin, csc cos, sec, tan,

cot III tan, cot sin, csc, cos,

secIV cos, sec sin, csc, tan, cot

5.2 - Trigonometric Function of Real Numbers

Example – pg. 384Find the exact value of the trigonometric functions at

the given real number.

5.2 - Trigonometric Function of Real Numbers

Examples – pg. 385Find the sign of the expression if the terminal point

determined by t is in the given quadrant.

tan sin49. , Quadrant III

cot

50. cos sec , any quadrant

t t

t

t t

5.2 - Trigonometric Function of Real Numbers

Even-Odd PropertiesSine, cosecant, tangent, and cotangent are odd

functions.Cosine and secant are even functions.

sin sin cos cos tan tan

csc csc sec sec cot cot

t t t t t t

t t t t t t

5.2 - Trigonometric Function of Real Numbers

Example – pg. 384Find the exact value of the trigonometric functions at

the given real number.

5.2 - Trigonometric Function of Real Numbers

Fundamental Identities

Reciprocal Identities

Pythagorean Identities

1 1csc sec

sin cos

sin 1 costan cot =

cos tan sin

t tt t

t tt t

t t t

2 2 2 2 2 2sin cos 1 tan 1 sec 1+cot csct t t t t t

5.2 - Trigonometric Function of Real Numbers

Examples – pg. 385Write the first expression in terms of the second if the

terminal point determined by t is in the given quadrant.

5.2 - Trigonometric Function of Real Numbers

Examples – pg. 385Find the values of the trigonometric functions of t

from the given information.

5.2 - Trigonometric Function of Real Numbers

Examples – pg. 384Find the exact value of the trigonometric functions at

the given real number.