Download Properties of the Trigonometric Functions

Properties of the
Trigonometric Functions
Domain and Range
• Remember:
•
sin   y
1
csc   y  0
y
cos   x
1
sec   x  0
x
y
tan   x  0
x
x
cot   y  0
y
Domain and Range
• The domain of the sine function is all real
numbers. The range of the sine function is
[-1, 1]
• The domain of the cosine function is all
real numbers. The range of the cosine
function is [-1, 1]
Domain and Range
• The domain of the tangent function is the
set of all real numbers, except odd multiples
of p/2. The range is all real numbers.
• The domain of the secant function is the set
of all real numbers, except odd multiples of
p/2. The range is (-∞, 1] u [1, ∞).
Domain and Range
• The domain of the cotangent function is the
set of all real numbers except integral
multiples of p. The range is all real numbers.
• The domain of the cosecant function is the
set of all real numbers except integral
multiples of p. The range is (-∞, 1] u [1, ∞)
Periodic Functions
• Definition:
• A function f is called periodic if there is a
positive number p such that, whenever θ
is in the domain of f, so is θ + p, and
• f(θ + p) = f(θ)
Periodic Properties
sin(  2p k )  sin 
csc(  2p k )  csc 
cos(  2p k )  cos 
sec(  2p k )  sec 
tan(  p k )  tan 
cot(  p k )  cot 
Periodic Functions
• If sin θ = 0.3, find the value of sin θ +
• sin (θ + 2p) + sin (θ + 4p)
• If tan θ = 3, find the value of tan θ +
• tan (θ + p) + tan (θ + 2p)
Signs of the Trigonometric
Functions
• Table 5 p. 403
• Remember the mnemonic (All – Quad I;
Scientists – Quad II; Take – Quad III;
Calculus – Quad IV)
Finding the Quadrant in Which an
Angle Lies
• If sin  < 0 and cos  < 0, name the
quadrant in which the angle lies.
• If sin  < 0 and tan  < 0, name the
quadrant in which the angle lies.
Fundamental Identities
• Reciprocal Identities:
1
csc  
sin 
1
sec  
cos 
Quotient Identities:
sin 
tan  
cos 
cos 
cot  
sin 
1
cot  
tan 
Fundamental Identities
• Pythagorean Identities:
sin   cos   1
2
2
tan   1  sec 
2
2
cot   1  csc 
2
2
Finding Exact Values of A Trig
Expression
5
2 5
Given sin  
and cos  
5
5
Find the other four trig functions using
identities and/or unit circle
Find the Exact Value of Trig
Functions
• Find the exact value of each expression.
Do not use a calculator.
a. sec 18  tan 18
cos 20
b. cot 20 
sin 20
c. tan 200 cot 20
25p
 p 
d . sin    csc
12
 12 
2
2
Given One Value of a Trig Function,
Find the Remaining Ones
• Given that tan θ=½ and sin θ < 0, find
the exact value of each of the remaining
five trig functions of θ.
• Using Definition
• Using Fundamental Identities
Even and Odd Properties
sin( )   sin 
csc( )   csc 
cos( )  cos 
sec( )  sec 
tan(  )   tan 
cot(  )   cot 
Properties of Trig Functions
• On-line Examples
• On-line Tutorial