|
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.
Question 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
Question 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.
Question 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)
Question 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
Question 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)
Question 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
Question 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)
More References on Trigonometry Questions
trigonometry questions with solutions and answers.
|