Download 2.2 Trigonometric Functions of Non-Acute Angles

2.2 Trigonometric Functions of Non‐Acute Angles Definition: Associated with every nonquadrantal angle in standard position is an acute angle, called its reference angle. The reference angle for a nonquadrantal angle in standard position is the acute angle between the x‐axis and the terminal side of . Examples: Find the reference angle for the given angle in standard position. 1. 1150 2. 2350 The reference angle is The reference angle is 180
115
65 235
180
55 3. 3100 4. 5170 The reference angle is The reference angle is 360
310
50 540
517
23 5. 11840 6. ‐14570 11840 = 3(3600)+1040 ‐11470 = 3(‐3600) ‐ 670 The reference angle 180
104
is The reference angle 76 is 670 Quadrant containing I II III IV Positive functions All sin and csc tan and cot cos and sec Negative functions None cos, sec, tan, and cot cos, sin, sec, and csc sin, csc, tan, and cot If is a nonquadrantal angle in standard position, then to find the value of a trigonometric function at , find its value for the reference angle and attach the appropriate sign. The appropriate sign can be determined from the table of signs of trigonometric functions from above. Examples: Find the six trigonometric function values of each angle. 7. 4800 4800 = 3600 + 1200 The reference angle is 180
120
60 sin 4800  sin 600 
csc 4800 
2 3
,
3
3
1
, cos 4800   cos 600   , tan 4800   tan 600   3
2
2
3
sec 4800  2,
cot 4800  
3
8. 5700 5700 = 3600 + 2100 The reference angle is 210
180
30 1
3
3
sin 5700   sin 300   , cos 5700   cos 300  
, tan 5700  tan 300 
2
2
3
csc 5700   csc 300  2,
9. 26700 26700 = 7(3600) + 1500 2 3
sec 5700   sec 300  
, cot 5700  cot 300  3
3
The reference angle is 180
150
30 1
3
3
sin 26700  sin 300   , cos 26700   cos 300  
, tan 26700   tan 300  
2
2
3
2 3
sec 5700   sec 300  
, cot 5700   cot 300   3
3
csc 5700  csc 300  2,
10. ‐4050 The reference angle is 405
360
45 

csc  405    csc 45
0
0






2
2
, cos 4050  cos 450 
, tan 4050   tan 450  1
2
2
0
0
0
0
  2, sec 405  sec 45  2, cot 405   cot 45  1
sin 4050   sin 450  

