Download Pre-Calculus Section 8.2: The Unit Circle and Its Functions

Pre-Calculus Section 8.2: The Unit Circle and Its Functions (Trigonometry)
1. Trigonometric Functions:
a:
b:
Let t be any real number and let P(x,y) be the
terminal point on the unit circle determined by t.
or
6. Example 1(E): Find the exact value of the
trigonometric function at the given real number.
or
or
7. Unit Circle:
2. Example 1(A): Find the value of each
a: sin(0)
d: sin( )
b: cos( )
e: tan ( )
The unit circle is a circle with radius 1 centered at
the origin in the xy plane. Its equation is defined
by the following:
c: tan( )
Note: Each (x,y) coordinate can also be defined by
(cos(t), sin(t)), where t = terminal point as
determined by the arc of the unit circle
3. Example 1(B): Find the value of each
a: sin( )
d: cos( )
b: cos( )
e: sin( )
Example: t =
produces the point
and
c: tan( )
4. Example 1(C): Find the value of all six trig
functions for each real number t given below:
a:
Additionally: Due to the sign values of (x,y) in
the coordinate plane, the value of (cos(t), sin(t))
will maintain the same sign as (x,y). Some people
remember sign values by using the saying, “All,
Scholars, Take, Calculus”
b:
5. Example 1(D): Find the value of all six trig
functions for each real number t given below:
8. Example 2(A): Find the sign of the expression if
the terminal point determined by t is in the given
quadrant.
Quadrant I
A: sin(t)
B: cos(t)
Quadrant II
C: tan(t)
D: sec(t)
12. Example 3 (C): Find the exact values of the
Quadrant III
E: csc(t)
trigonometric functions
and
.
F: cot(t)
Quadrant IV:
G: sin(t)
13. Example 3 (D): Find the exact values of the
H: sec(t)
trigonometric functions
9. Example 2 (B): From the information given, find
the quadrant in which the terminal point determined
by t lies.
and
.
14. Example 3 (E): Find the exact values of the
trigonometric functions
and
.
and
15. Example 3 (F): Find the exact values of the
10. Example 3 (A): Find the exact values of the
trigonometric functions
trigonometric functions
and
and
16. Example 3 (G): Find the exact values of the
a.
trigonometric functions
and
b.
c.
17. Example 3 (H): Find the exact values of the
trigonometric functions
11. Example 3 (B): Find the exact values of the
trigonometric functions
and
and
.
18. Example 3 (I): Find the exact values of the
trigonometric functions
and
19. Example 3 (J): Find the exact values of the
trigonometric functions
and
.
(a) By using the figure, estimate
(b) By using a calculator in radians find
20. Example 3(K): Find the value of each
a: sin(
)
b: cos(
c: tan(
d: cos(
)
e: sin(
)
)
23. Example 4 (C): Find the approximate value of
)
21. Example 4 (A): Find the approximate value of
(a) By using the figure. Please give the answer to
one decimal place.
__________
(a) By using the figure find:
(b) By using a calculator in radians find:
22. Example 4 (B): Find the approximate value of
(b) By using a calculator in radians. Please give the
answer to five decimal places.
__________
24. Even-Odd Properties:
Sine, Cosecant, Tangent, and Cotangent are Odd
Functions:
sin(-t) = -sin(t)
csc(-t) = -csc(t)
25. Example 5 (A): Determine whether the function
tan(-t) = -tan(t)
cot(-t) = -cot(t)
is even, odd, or neither.
Cosine and Secant are Even Functions:
a. neither
b. odd
c. even
Example 5 (B): Determine whether each function is even, odd, or neither. Match each function with the
corresponding concept.
cos(-t) = cos(t)
a.
sec(-t) = sec(t)
c.
b.
26. even
28. neither
27. odd
29. Example 6 (A): Find the values of the trigonometric functions of t if
and the terminal point of t is in
quadrant II.
30. Example 6 (B): Find the values of the trigonometric functions of t if
and the terminal point of t is in
quadrant III.
31. Example 6 (C): Find the values of the trigonometric functions of t if
32. Example 6 (E): Find the values of the trigonometric functions of t if
in quadrant IV.
33. Pythagorean Identities:
and
.
and the terminal point of t is
34. Example 7(A): Write sin(t) in terms of cos(t),
where t is in Quadrant II.
35. Example 7(B): Write cos(t) in terms of sin(t), where
t is acute.
36. Example 7(C): Write tan(t) in terms of cos(t),
where t is in Quadrant III.
37. Example 7(D): Write sec( ) in terms of cos( ),
where is in Quadrant IV.
38. Example 7(E): Write sec( ) in terms of sin( ),
where is in Quadrant IV.
39. Example 7(F): Write tan( ) in terms of sin( ),
where is in Quadrant III.
40. Example 7(G): Write sin( ) in terms of cot( ),
where is in Quadrant IV.