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Reciprocal functions secant, cosecant, cotangent sec secant csc cosecant cot cotangent Hyp Opp sin Hyp Adj cos Adj Opp tan Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Adj Hyp sec Cosecant is the reciprocal of sine. Reciprocal means to flip the ratio. Opp Hyp csc Cotangent is the reciprocal of tangent. Reciprocal means to flip the ratio. Opp Adj cot

Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of

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Page 1: Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of

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

Page 2: Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of

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

Page 3: Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of

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!

Page 4: Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of

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!

Page 5: Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of

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!

Page 6: Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of

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!

Page 7: Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of

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

Page 8: Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of

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 9: Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of

Page 10

150sec

150

30150180

30

30cos30sec

30cos30sec

2

330sec

Sec is the reciprocal of cos, so:

2

3

3

2 1

Page 10: Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of

Page 10

1791.1238csc

179.1238csc

Page 11: Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of

Page 10

4281.135cot

428.135cot

Page 12: Reciprocal functions secant, cosecant, cotangent Secant is the reciprocal of cosine. Reciprocal means to flip the ratio. Cosecant is the reciprocal of

Page 10

xxf sec2

30sec230f

2

330cos

3

230sec

3

2230sec2

3

430sec2

3

3

3

4

9

34

3

34

3

3430sec2