Transcript 4.4 Trigonmetric functions of Any Angle

4.4 Trigonmetric functions of Any
Angle
Objective
• Evaluate trigonometric functions of any
angle
• Use reference angles to evaluate trig
functions
Definitions of Trigonometric
Functions of any Angle
• Let θ be an angle in standard position with (x, y)
a point on the terminal side of θ and
r x y 0
2
2
y
x
sin  
cos  
r
r
y
x
tan   , x  0 cot   y  0
x
y,
r
r
sec   , x  0 csc   , y  0
x
y
• The cosecant function is the reciprocal of
the sine.
• The secant function is the reciprocal of the
cosine.
• The cotangent function is the reciprocal of
the tangent function.
Example 1
• Let (-3, 4) be a point
on the terminal side of
θ. Find the sine,
cosine, and tangent of
θ.
r  x2  y 2
r  (3) 2  42
r  9  16
r 5
y 4
sin   
r 5
x
3
cos    
r
5
y
4
tan    
x
3
Example 2
• Let (2, 5) be a
point on the
terminal side of θ.
Find the sine,
cosine, and
tangent of θ.
r  x2  y2
r  (2) 2  52
r  4  25
r  29
y
5
5
29 5 29
sin   



r
29
29
29 29
x
4
4
29 4 29




r
29
29
29 29
y 5
tan   
x 4
cos  
Signs of the Trigonometric
Functions
Signs of the Trig Functions
A means that all trig. functions are positive.
S means that all sine and cosecant functions are positive.
T means that all tangent and cotangent functions are positive.
C means that all cosine and secant functions are positive.
Example 3
• State whether each value is positive,
negative, or zero.
• a) cos 75° positive
• b) sin 3π 0
• c) cos 5π negative
• d) sin(-3π) 0
Example 4
• Given.
4
and tan   0, find cos  and csc  .
5
4 y
sin    , implies y = 4 and r = 5
5 r
since tan <0, and y = 4,  is in the II quadrant
sin  
r  x2  y 2
5  x 2  42
25  x 2  16
9  x2
x  3, since  is in II, x = -3
x
3
r 5
cos    , csc   
r
5
y 4
Example 5
• Angle θ is in
standard position
with its terminal side
in the third
quadrant. Find the
exact value of cos θ
if
1
2
1 y
sin     ,implies y = -1, r = 2
2 r
sin   
r  x2  y 2
2  x 2  (1) 2
4  x2  1
3  x 2 , x   3,since  is in III, x   3
x
3
cos    
r
2
Example 6
• Angle θ is in standard
position with its
terminal side in the
fourth quadrant. Find
the exact value of sin
θ if
4
7
4 x
cos    ,implies x = 4, r = 7
7 r
cos  
r  x2  y 2
7  42  y 2
49  16  y 2
33  y 2 , y   33,since  is in IV, y   33
sin  
y
33

r
7
Reference Angles
• Definition
• Let θ be an angle in standard position. Its
reference angle is the acute angle θ’
formed by the terminal side of θ and the
horizontal axis.
Reference angles
Example 7
• Finding reference angles.
a.   213
b.   1.7
c.   144
Trigonometric Values of
Common Angles
Example 8
• Use the reference angle to find sin θ,
and tan θ for each value of
a. 150 is in II so  '  180  150  30
1
3
1
3
sin 30  , cos 30 
implies sin150  , cos  
,
2
2
2
2
1
3
tan150  1/ 2


 3/2
3
3
b. 330 is in IV so  '  360  330  30
1
3
3
sin 330   , cos 330 
, tan 330  
2
2
3
7
7

c.
is in III, so  ' 
    30
6
6
6
sin
7
1
7
3
7
3
  , cos

, tan

6
2
6
2
6
3
cos θ,
Example 9
For 0    2 ,
• Determine the values of θ for which
1
sin   , looking at the unit circle
2

5
  , 
6
6
• If the value of one of the trig functions of
any angle is known, a calculator can be
used to determine the angles having that
value.
Example 10
• Find values of θ, where
0 
• to the nearest tenth of a degree.
a. cos   .9266
Make sure calculator is in degrees
2nd cos(.9266) = 22.1
b. sin   0.6009
2nd sin(-0.6009)  36.9
c. tan   .2309
2nd tan(.2309)  13
  360
Example 11
0    2
• Find values of θ, where
• To the nearest hundredth of a radian.
a. tan   3.009
Make sure calculator is in radians
2nd tan(3.009)  1.25 radians
b. cot   4.69
2nd tan( 1/ 4.69)  .21 radians
c. sec   8.2986
3nd cos(1/8.2986)  1.44 radians