
Let us assume that we want to find the exact value of f(x), where f is any of the six trigonometric functions sine, cosine, tangent, cotangent, secant and cosecant. To find the exact value of f(x), we suggest the following steps:
1  If the angle x is negative, we first use a formula for negative angles such as sin ( x) =  sin (x), cos ( x) = cos (x) and so on.
2  Next we locate the terminal side of the angle in question, directly or using a positive coterminal angle t, which gives the sign of the trigonometric function.
3  We find the reference angle Tr to the angle in question and use that fact that f(x) = + or  f(Tr). The sign + or  is determined using the quadrant found in step 2. If the angle in question or its coterminal angle are in quadrant 1, this last step is not needed.
Questions 1:
Find the exact value of sin ( Pi / 3).
Solution to Question 1:
 Use the identity for negative angles, to write
sin ( Pi / 3) =  sin (Pi / 3)
 Pi / 3 is in quadrant 1 and there is no need for either coterminal or reference angles. sin ( Pi / 3) is evaluated directly as follows
sin ( Pi / 3) =  sin (Pi / 3) =  sqrt (3) / 2
Questions 2:
Find the exact value of cos ( 390 ^{o}).
Solution to Question 2:
 We use the identity cos(x) = cos(x) to write
cos ( 390 ^{o}) = cos (390 ^{o})
 Since 390 ^{o} is greater than 360 ^{o}, we find a coterminal angle t, greater than zero and less than 360 ^{o}, to 390 ^{o}.
t = 390  (360) = 30 ^{o}
 Note that since 390 ^{o} and angle t = 30 are coterminal, we can write
cos (390 ^{o}) = cos ( 30 ^{o} )
 Finally.
cos( 330 ^{o}) = cos ( 330 ^{o} )
= cos (30) = sqrt (3) / 2
 Note that there was no need for reference angle since 30 ^{o} is in quadrant 1.
Questions 3:
Find the exact value of sec (3 Pi / 4).
Solution to Question 3:
 3 Pi / 4 has its terminal side in quadrant 2. In quadrant 2 the secant is negative. Hence
sec (3 Pi / 4) =  sec(Tr)
 where Tr is the reference angle to 3 Pi / 4 and is given by
Tr = Pi  3 Pi / 4 = Pi / 4
 Hence
sec (3 Pi / 4) =  sec(Pi / 4) =  sqrt(2)
Questions 4:
Find the exact value of cot ( 840 ^{o}).
Solution to Question 4:
 840 ^{o} is positive and greater than 360 ^{o}, hence the need to first find the coterminal angle t
t = 840 ^{o}  2 (360) ^{o} = 120 ^{o}
 We can write
cot ( 840 ^{o}) = cot (120 ^{o})
 120 ^{o} is in quadrant 2 where the cotangent is negative, hence
cot ( 840 ^{o}) = cot (120 ^{o}) =  cot (Tr)
 120 ^{o} is in quadrant 2, hence its reference angle Tr is given by
Tr = 180  120 = 60 ^{o}
 Finally
cot ( 840 ^{o}) =  cot (60 ^{o}) =  sqrt(3) / 3
Questions 5:
Find the exact value of csc ( 7 Pi / 4).
Solution to Question 5:
 Negative angle identity gives
csc ( 7 Pi / 4) =  csc ( 7 Pi / 4 )
 The terminal angle of 7 Pi / 4 is in quadrant 4 where the cosecant is negative. The reference Tr angle of 7 Pi / 4 is given by
Tr = 2 Pi  7 Pi / 4 = Pi / 4
 Hence
csc ( 7 Pi / 4 ) =  csc (Pi / 4) =  sqrt(2)
 Finally, substitute the above into csc ( 7 Pi / 4) =  csc ( 7 Pi / 4 ) to obtain
csc ( 7 Pi / 4) = sqrt(2)
Questions 6:
Find the exact value of cot (121 Pi / 3).
Solution to Question 6:
 We first note that
121 Pi / 3 = 120 Pi / 3 + Pi / 3
= 40 Pi + Pi / 3
 A positive coterminal angle t to 121 Pi / 3 may be calculated as follows
t = 121 Pi / 3  20 (2 PI) = 121 Pi / 3  40 Pi = Pi / 3
 The coterminal angle is in quadrant 1 and there is no need for the reference angle. Hence
cot (121 Pi / 3) = cot (Pi / 3) = sqrt (3) / 3
Questions 7:
Find the exact value of sec (  3810 ^{o}).
Solution to Question 7:
 Use negative angle identity
sec (  3810 ^{o}) = sec (3810 ^{o})
 Note that
3810 ^{o} = 3600 ^{o} + 210 ^{o}
 The coterminal angle t to 3810 ^{o} may be calculated as follows
t = 3810 ^{o}  10(360) ^{o} = 210 ^{o}
 The terminal side of the coterminal angle t is in quadrant 3 and where therefore sec (3810 ^{o}) is negative and given by
sec (3810 ^{o}) =  sec(Tr)
 where Tr is the reference angle to angle 210 ^{o} and is given by
Tr = 210 ^{o}  180 ^{o} = 30 ^{o}
 Hence
sec (  3810 ^{o}) =  sec (30 ^{o}) =  2 / sqrt(3)
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