Download Section 5.2анаTrigonometric Functions of Real Numbers Definition

Section 5.2 ­ Trigonometric Functions of Real Numbers
Definition of the Trig Functions ­ Let t be any real number and let P(x, y) be the terminal point on the unit circle determined by t. Then:
sin t = y cos t = x tan t = (x ≠0)
y
x
x
1
1
y
y
csc t = (y
x ≠0)cot t = (y ≠0)
≠0) sec t = (x
Oct 5­7:24 AM
1
sinx
cosx
sinx
cosx
tanx
tanx
sinx
0
cscx
cscx
secx
secx
cotx
cotx
Nov 30­8:48 AM
2
Domains of the Trig Functions
FunctionDomain
sin, cos All real numbers
tan, sec All real numbers other than + n π π
2
cot, csc All real numbers other than n π
Signs of the Trig Functions
Oct 8­3:05 PM
3
Even­Odd Properties ­­ sin, csc, tan and cot are odd functions; cos and sec are even functions
sin(­t) = ­sin t cos(­t) = cos t
csc(­t) = ­ csc t sec(­t) = sec t
tan(­t) = ­tan t
cot(­t) = ­cot t
Oct 9­6:38 AM
4
Find the sign of the expression if the terminal point determined by t is in the given quadrant.
tan(t)csc(t) in quadrant II
From info. given, find the quadrant in which the terminal point determined by t lies: cos (t) < 0 and cot (t) < 0 Dec 5­8:33 AM
5
Fundamental Identities
Reciprocal Identities
csc t =
sec t = 1
cos t
1
sin t
sin t
tan t = cos t
cot t = cot t = 1
tan t
cos t
sin t
Pythagorean Identities
sin2 t + cos2 t = 1
tan2 t + 1 = sec 2 t
1 + cot2 t = csc2 t
Oct 9­6:41 AM
6
Using the calculator to find trig values
Using
Remember: 1. If the rotation is given in radians your calculator must be set to radian mode
2. To find the csc, sec or cot you must
calculate ________1_______
the appropriate reciprocal function
Dec 3­7:08 AM
7