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Reciprocal functionssecant, cosecant, cotangent
secsecant csccosecant cotcotangent
Hyp
Oppsin
Hyp
Adjcos
Adj
Opptan
Secant is the reciprocal of cosine. Reciprocal means to flip the ratio.
Adj
Hypsec
Cosecant is the reciprocal of sine. Reciprocal means to flip the ratio.
Opp
Hypcsc
Cotangent is the reciprocal of tangent. Reciprocal means to flip the ratio.
Opp
Adjcot
Sec, csc & cot also have the same SIGN as their reciprocals.
AS
T C
Now all six functions are positive
Just sin and csc positive
Just tan and cot positive
Just cos and sec positive
Determine the quadrant in which x terminates:
0tan and 0sec .1 xAS
T C
Now all six functions are positive
Just sin and csc positive
Just tan and cot positive
Just cos and sec positive
negative means 0sec x
Since sec is the reciprocal function of cos, then sec and cos have the same SIGN in the quadrants. If sec is negative, so is cos. Thus we are in quadrants II and III.
Tangent is positive, so we are in quadrants I and III.
x
x
x
x
Since they overlap in quadrant III, then that is where x terminates!
Determine the quadrant in which x terminates:
0cot and 0csc .2 xxAS
T C
Now all six functions are positive
Just sin and csc positive
Just tan and cot positive
Just cos and sec positive
Think, where is sin positive?
xx
Think, where is tan negative?
x
x
Since they overlap in quadrant II, then that is where x terminates!
Determine the quadrant in which x terminates:
0sec and 0sin .3 xxAS
T C
Now all six functions are positive
Just sin and csc positive
Just tan and cot positive
Just cos and sec positive
Think, where is sin negative?
xx
Think, where is cos negative?
x
x
Since they overlap in quadrant III, then that is where x terminates!
Find the exact value of each expression:
Step:
1. Start by drawing the given angle
2. Find the reference angle
3. Rewrite the function using reference angle
4. Determine SIGN of function in quadrant where it was drawn
5. Now find exact value using exact value chart (use reciprocal function to get value off chart)
6. Take reciprocal of value off chart to get final answer!
120sec
120
60120180
60
60sec
60cos
2
1
Cos in quad II is negative
Use cos because sec is not on exact value chart and cos is reciprocal of sec!
60cosSince cos is negative in II, sec is as well
60sec 2When you switch back to sec, you take reciprocal of exact value!
This is how you can check your answer on the calculator!
Find the exact value of each expression:
300cot
300
60300360 60
60cot60tan 60tan 3
60cot3
1 We have a
problem here!
3
3
3
1
9
3
3
3
60cot3
3
Find the exact value of each expression:
225csc
225
45180225 45
45csc45sin
45sin2
2
45csc2
2
2
2
2
2
4
22
2
22
2
45csc 2
Page 10
150sec
150
30150180
30
30cos30sec
30cos30sec
2
330sec
Sec is the reciprocal of cos, so:
2
3
3
2 1
Page 10
1791.1238csc
179.1238csc
Page 10
4281.135cot
428.135cot
Page 10
xxf sec2
30sec230f
2
330cos
3
230sec
3
2230sec2
3
430sec2
3
3
3
4
9
34
3
34
3
3430sec2