Download Lecture Notes for Section 5.2

Math Analysis Notes
Section 5.2
Page 1 of 3
Section 5.2: Trigonometric Functions of Real Numbers
Big Idea: The x and y coordinates of a terminal point on the unit circle are defined by six trigonometric
functions.
Big Skill: You should be able to find the exact value of the trig functions at the special angles talked about
in class, and be able to calculate the value of any trig function given the value of one trig function.
1. The Six Trig Functions defined on the Unit Circle:
t is the angle to the terminal point; (x, y) is the coordinate of the terminal point.
opp.
a. sin t  y ;
;
domain: {t | t  }
sin t 
hyp.
adj.
b. cost  x ;
;
domain: {t | t  }
cos t 
hyp.
y

opp.
c. tan t  ;
;
domain: {t | t 
and t  + n for integer n}
tan t 
x
2
adj.
adj.
x
d. cot t  ;
;
domain: {t | t  and t  n for integer n}
cot t 
opp.
y
1

hyp.
e. sec t  ;
;
domain: {t | t 
and t  + n for integer n}
sec t 
x
2
adj.
hyp.
1
f. csct  ;
;
domain: {t | t 
and t  n for integer n}
csc t 
opp.
y
Math Analysis Notes
t
0

6

4

3

2
2
3
3
4
5
6

7
6
5
4
4
3
3
2
5
3
7
4
11
6
2
sin(t)
Section 5.2
cos(t)
tan(t)
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cot(t)
sec(t)
csc(t)
Math Analysis Notes
Section 5.2
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2. The signs of the six trig functions are determined by which of the four quadrants the terminal point is in.
3. Even-Odd Properties:
a. sin(-t) = -sin(t) (an odd function)
b. cos(-t) = cos(t) (an even function)
c. tan(-t) = -tan(t) (an odd function)
d. cot(-t) = -cot(t) (an odd function)
e. sec(-t) = sec(t) (an even function)
f. csc(-t) = -csc(t) (an odd function)
4. Reciprocal Identities:
1
a. csc t 
sin t
1
b. sec t 
cos t
1
cos t

c. cot t 
tan t sin t
1
sin t

d. tan t 
cot t cos t
e. Note: you need these identities on a calculator, since most calculators don’t have these functions
built in.
5. Pythagorean Identities:
a. sin 2 t  cos 2 t  1
b. tan 2 t  1  sec 2 t
c. cot 2 t  1  csc2 t
Practice:
 3 4
1. Find the six trig functions for a terminal point of   ,  .
 5 5
2. Use the unit circle to estimate cos 2.5.
3. Find the other five trig functions of t when sin t  
1
and the terminal point is in the third quadrant.
3
4. Find the other five trig functions in terms of sin t when the terminal point is in the third quadrant.