properties of the trigonometric functions

Properties of the Properties of the Trigonometric FunctionsTrigonometric Functions

Domain and RangeDomain and Range

• Remember:Remember:

• 1sin csc 0

1cos sec 0

tan 0 cot 0

y yy

x xx

y xx y

x y

Domain and RangeDomain and Range

• The domain of the sine function is all The domain of the sine function is all real numbers. The range of the sine real numbers. The range of the sine function is [-1, 1]function is [-1, 1]

• The domain of the cosine function is The domain of the cosine function is all real numbers. The range of the all real numbers. The range of the cosine function is [-1, 1]cosine function is [-1, 1]

Domain and RangeDomain and Range

• The domain of the tangent function is the The domain of the tangent function is the set of all real numbers, except odd set of all real numbers, except odd multiples of multiples of /2. The range is all real /2. The range is all real numbers.numbers.

• The domain of the secant function is the set The domain of the secant function is the set of all real numbers, except odd multiples of of all real numbers, except odd multiples of /2. The range is (-∞, 1] u [1, ∞). /2. The range is (-∞, 1] u [1, ∞).

Domain and RangeDomain and Range

• The domain of the cotangent function is The domain of the cotangent function is the set of all real numbers except integral the set of all real numbers except integral multiples of multiples of . The range is all real . The range is all real numbers.numbers.

• The domain of the cosecant function is The domain of the cosecant function is the set of all real numbers except integral the set of all real numbers except integral multiples of multiples of . The range is . The range is (-∞, 1] u [1, ∞) (-∞, 1] u [1, ∞)

Periodic FunctionsPeriodic Functions

• Definition:Definition:

• A function A function ff is called periodic if there is called periodic if there is a positive number is a positive number pp such that, such that, whenever whenever θθ is in the domain of is in the domain of ff, so , so is is θθ + + pp, and, and

• f(f(θθ + + p) = f(p) = f(θθ))

Periodic PropertiesPeriodic Properties

sin( 2 ) sin cos( 2 ) cos

csc( 2 ) csc sec( 2 ) sec

tan( ) tan cot( ) cot

k k

k k

k k

Periodic FunctionsPeriodic Functions

• If sin If sin θθ = 0.3, find the value of sin = 0.3, find the value of sin θθ ++

• sin (sin (θθ + 2 + 2) + sin () + sin (θθ + 4 + 4))

• If tan If tan θθ = 3, find the value of tan = 3, find the value of tan θθ + +

• tan (tan (θθ + + ) + tan () + tan (θθ + 2 + 2))

Signs of the Trigonometric Signs of the Trigonometric FunctionsFunctions

• Table 5 p. 403Table 5 p. 403

• Remember the mnemonic (All – Quad Remember the mnemonic (All – Quad I; Scientists – Quad II; Take – Quad III; I; Scientists – Quad II; Take – Quad III; Calculus – Quad IV)Calculus – Quad IV)

Finding the Quadrant in Which Finding the Quadrant in Which an Angle Liesan Angle Lies

• If sin If sin and cos and cos < 0, name the < 0, name the quadrant in which the angle lies.quadrant in which the angle lies.

• If sin If sin < 0 and tan < 0 and tan < 0, name the < 0, name the quadrant in which the angle lies.quadrant in which the angle lies.

Fundamental IdentitiesFundamental Identities

• Reciprocal Reciprocal Identities:Identities:

Quotient Identities: Quotient Identities:

1 1 1csc sec cot

sin cos tan

sin costan cot

cos sin

Fundamental IdentitiesFundamental Identities

• Pythagorean Identities:Pythagorean Identities:

2 2

2 2

2 2

sin cos 1

tan 1 sec

cot 1 csc

Finding Exact Values of A Trig Finding Exact Values of A Trig ExpressionExpression

5 2 5sin cos

5 5Given and

Find the other four trig functions using identities and/or unit circle

Find the Exact Value of Trig Find the Exact Value of Trig FunctionsFunctions

• Find the exact value of each Find the exact value of each expression. Do not use a calculator.expression. Do not use a calculator.

2 2 cos 20. sec 18 tan 18 . cot 20

sin 20

25. tan 200 cot 20 . sin csc

12 12

a b

c d

Given One Value of a Trig Given One Value of a Trig Function, Find the Remaining Function, Find the Remaining Ones Ones • Given that tan Given that tan θθ=½ and sin =½ and sin θθ < 0, < 0,

find the exact value of each of the find the exact value of each of the remaining five trig functions of remaining five trig functions of θθ..

• Using DefinitionUsing Definition

• Using Fundamental IdentitiesUsing Fundamental Identities

Even and Odd PropertiesEven and Odd Properties

sin( ) sin cos( ) cos tan( ) tan

csc( ) csc sec( ) sec cot( ) cot

Properties of Trig FunctionsProperties of Trig Functions

• On-line ExamplesOn-line Examples

• On-line TutorialOn-line Tutorial