Objectives:
Evaluate trigonometric functions of any angle
 In this section, the definitions are extended to cover
any angle.
 Let  be an angle in standard position with  x, y  a
point on the terminal side of  and r  x 2  y 2  0
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
 In the definition of the trigonometric functions, r is
the distance from the origin to the point  x, y  .
Distance is never negative, so r  0 . The location of the
angle  determines the sign of the trig function, since
the quadrant will tell us if x and y are positive or
negative.

 1.
 3, 4 
 2.
 4, 1
 3.
5
Given tan    and  is in Quad IV
4
Reciprocal Identities
 Fundamental Identities:
1
1
sin  
csc 
Pythagorean Identities
csc
sin 
1
1
sin 2   cos 2   1
cos 
sec 
sec
cos
2
2
1  tan   sec 
1
1
2
2
tan  
cot  
cot   1  csc 
cot 
tan 
Quotient Identities Cofunction Identities
For any acute angle A
sin 
tan  
sin A  cos  90  A  cos A  sin  90  A 
cos
cos
tan A  cot  90  A  cot A  tan  90  A 
cot  
sin 
sec A  csc  90  A  csc A  sec  90  A 
 4. Given sin  
1
and  is in Quad II
3
 5.
15
Given cos   
and  is in Quad III
17
 6. Given sin  
4
and tan   0
5
 With an angle  drawn on a coordinate plane, the
right triangle formed has certain distance values. The r
will always be the longest side or distance.
y  r and x  r
y
x
1   1 and  1   1
r
r
1  sin   1 and  1  cos  1
 The functions sec and csc are reciprocals of the
functions cos and sin  , respectively.
sec  1 or sec  1 and csc  1 or csc  1
 Therefore sec and csc are never between -1 and 1
y
 The tangent of an angle is defined as tan   . It is
x
possible that x  y, x  y, or x  y . For this reason y
can take on any value at all, so tan  and cot  can x
be any real number.
 EX 7: Decide whether the following statements are
possible or impossible. Explain.
 a) sin   8
b) tan   110.47
 c)
sec   0.6
 It can be shown from the right triangle definitions that
cofunctions of complementary angles are equal. That
is, if  is an acute angle, then the cofunction identities
are true.
 EX 8: Write each of the following in terms in
Cofunctions.
 a) cos52
 c)
4
sec
5
b) tan 71
d) sin

12