2.5 Properties of the Trig Functions
Warm-up (IN) – none!
Learning Objective: To find the exact values of the Trig functions using the unit circle, to know the domain and range of the trig functions, and to use the Periodic and Even-Odd Properties to find the Exact values of the Trig functions.
If t is a real # and P is any point on the unit circle...
( , )P a b sin t b
cos t a
tan t b
a
csc t 1
b
sec t 1
a
cot t a
b
t
0a
0b
0a
0b
NOTES
Ex 1 – Find all 6 trig functions if P = is the point on the unit circle that corresponds to t.
1 3,
2 2
sin t 3
2
cos t 1
2
tan t 3
csc t 2 3
3
sec t 2
cot t 3
3
If P is any point on circle and is on the terminal side of …
2 2 2x y r
sin t b
r
cos t a
r
tan t b
a
csc t r
b
sec t r
a
cot t a
b
2 2r x y
Ex 2 – Find all 6 trig functions if P = 2, 4
sin t 4
2 5
cos t 2
2 5
tan t 4
2
csc t 5
2
sec t 5
cot t 1
2
222 4r
20r
2 5r 2 5
5
5
5
2
Periodic Properties
periodic – means repeating
sin, cos, csc and sec are the same every360º or 2π radians
tan and cot are the same every180º or π radians
Ex 3 – Find the exact value:
So… sin 2 sink
cos 2 cosk
tan tank
k meaning any multiple of 2π
a. sin 420
5b. tan
4
4tan
4 4
tan4
1
11c. cos
4
3 8cos
4 4
3
cos4
2
2
)60360sin( 60sin 2
3
D:
R:
Domain and Range of 6 trig functions
sin cos tan csc sec cot
( , )P a bgiven
b a b
a1
b1
a
a
b
Except odd multiples of
2
Except odd multiples of
Except odd multiples of
2
Except odd multiples of
1 1y 1 1y
Except odd multiples of
2
1
and
1
y
y
1
and
1
y
y
Except odd multiples of
Even-Odd Functions
sin -sin
cos cos
tan - tan
csc -csc
sec sec
cot cot
Odd
Even
Odd
Ex 4 – Find the exact value:
a. sin 45 sin 45 2
2
b. cos cos 13
c. cot2
3
cot2
0
37d. tan
4
37
tan4
1
HW – pg. 172 #13-17,21-24,29-33,37-41, 47-51,55-57,61-64, 67-72
Out – What does periodic mean?
Summary - Today was …
Don’t forget
about POW!!
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