For more information, please see full course syllabus of Trigonometry

For more information, please see full course syllabus of Trigonometry

### Secant and Cosecant Functions

**Main definitions**:

- The
*secant function*is defined by

for values of θ where cosθ ≠ 0. (For values of θ where cosθ = 0, the secant is undefined.)secθ = 1 cosθ - The
*cosecant function*is defined by

for values of θ where sinθ ≠ 0. (For values of θ where sinθ = 0, the cosecant is undefined.)cscθ = 1 sinθ

**Example 1**:

**Example 2**:

**Example 3**:

**Example 4**:

**Example 5**:

^{R}, 240

^{° }, and (7π/4)

^{R}.

### Secant and Cosecant Functions

^{°}

- Find the sine and cosine of 120
^{°}. Note that 120^{°}= [(2π)/3] - Locate 120
^{°}on the unit circle. - We know that cosine is the x coordinate of 120
^{°}and sine is the y coordinate of 120^{°} - sin120
^{°}= [(√3 )/2], and cos120^{°}= − [1/2] - cscθ = [1/(sinθ)], and secθ = [1/(cosθ)]

^{°}= [1/([(√3 )/2])] = [(2√3 )/3], sec120

^{°}= [1/( − [1/2])] = − 2

- Find the sine and cosine of [(11π)/6]. Note that [(11π)/6] = 330
^{°} - Locate [(11π)/6] on the unit circle.
- We know that cosine is the x coordinate of [(11π)/6] and sine is the y coordinate of [(11π)/6]
- sin[(11π)/6] = − [1/2], and cos[(11π)/6] = [(√3 )/2]
- cscθ = [1/(sinθ)], and secθ = [1/(cosθ)]

^{°}

- Find the sine and cosine of 225
^{°}. Note that 225^{°}= [(5π)/4] - Locate 225
^{°}on the unit circle. - We know that cosine is the x coordinate of 225
^{°}and sine is the y coordinate of 225^{°} - sin225
^{°}= − [(√2 )/2], and cos225^{°}= − [(√2 )/2] - cscθ = [1/(sinθ)], and secθ = [1/(cosθ)]

^{°}= − √2, sec225

^{°}= − √2

- Find the sine and cosine of [(3π)/4]. Note that [(3π)/4] = 135
^{°} - Locate [(3π)/4] on the unit circle.
- We know that cosine is the x coordinate of [(3π)/4] and sine is the y coordinate of [(3π)/4]
- sin[(3π)/4] = [(√2 )/2], and cos[(3π)/4] = − [(√2 )/2]
- cscθ = [1/(sinθ)], and secθ = [1/(cosθ)]

^{°}

- Find the sine and cosine of 300
^{°}. Note that 300^{°}= [(5π)/3] - Locate 300
^{°}on the unit circle. - We know that cosine is the x coordinate of 300
^{°}and sine is the y coordinate of 300^{°} - sin300
^{°}= − [(√3 )/2], and cos300^{°}= [1/2] - cscθ = [1/(sinθ)], and secθ = [1/(cosθ)]

^{°}= − [(2√3 )/3], sec300

^{°}= 2

- Find the sine and cosine of [(7π)/6]. Note that [(7π)/6] = 210
^{°} - Locate [(7π)/6] on the unit circle.
- We know that cosine is the x coordinate of [(7π)/6] and sine is the y coordinate of [(7π)/6]
- sin[(7π)/6] = − [1/2], and cos[(7π)/6] = − [(√3 )/2]
- cscθ = [1/(sinθ)], and secθ = [1/(cosθ)]

- Start by graphing f(x) = cos[x/2]. Calculate the period by solving [x/2] = 0 and [x/2] = 2π. x = 0 and x = 4π
- The zeroes of f(x) = cos[x/2] are the vertical asymptotes of f(x) = sec[x/2]. Now graph f(x) = sec[x/2].

- Start by graphing f(x) = sin2x. Calculate the period by solving 2x = 0 and 2x = 2π. x = 0 and x = π
- The zeroes of f(x) = sin2x are the vertical asymptotes of f(x) = csc2x. Now graph f(x) = csc2x.

- Start by graphing f(x) = - 2cos2x. Calculate the period by solving 2x = 0 and 2x = 2π. x = 0 and x = π
- The zeroes of f(x) = - 2cos2x are the vertical asymptotes of f(x) = - 2sec2x. Now graph f(x) = - 2sec2x.

- Start by graphing f(x) = − [1/2]sin[x/2]. Calculate the period by solving [x/2] = 0 and [x/2] = 2π. x = 0 and x = 4π
- The zeroes of f(x) = − [1/2]sin[x/2] are the vertical asymptotes of f(x) = − [1/2]csc[x/2]. Now graph f(x) = − [1/2]csc[x/2].

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Secant and Cosecant Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Secant and Cosecant Definitions 0:17
- Secant Definition
- Cosecant Definition
- Example 1: Graph Secant Function 0:48
- Example 2: Values of Secant and Cosecant 6:49
- Example 3: Odd, Even, or Neither 12:49
- Extra Example 1: Graph of Cosecant Function
- Extra Example 2: Values of Secant and Cosecant

### Trigonometry Online Course

I. Trigonometric Functions | ||
---|---|---|

Angles | 39:05 | |

Sine and Cosine Functions | 43:16 | |

Sine and Cosine Values of Special Angles | 33:05 | |

Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D | 52:03 | |

Tangent and Cotangent Functions | 36:04 | |

Secant and Cosecant Functions | 27:18 | |

Inverse Trigonometric Functions | 32:58 | |

Computations of Inverse Trigonometric Functions | 31:08 | |

II. Trigonometric Identities | ||

Pythagorean Identity | 19:11 | |

Identity Tan(squared)x+1=Sec(squared)x | 23:16 | |

Addition and Subtraction Formulas | 52:52 | |

Double Angle Formulas | 29:05 | |

Half-Angle Formulas | 43:55 | |

III. Applications of Trigonometry | ||

Trigonometry in Right Angles | 25:43 | |

Law of Sines | 56:40 | |

Law of Cosines | 49:05 | |

Finding the Area of a Triangle | 27:37 | |

Word Problems and Applications of Trigonometry | 34:25 | |

Vectors | 46:42 | |

IV. Complex Numbers and Polar Coordinates | ||

Polar Coordinates | 1:07:35 | |

Complex Numbers | 35:59 | |

Polar Form of Complex Numbers | 40:43 | |

DeMoivre's Theorem | 57:37 |

### Transcription: Secant and Cosecant Functions

*We are trying some more examples for the sec and cosec function.*0000

*Here we are being asked to draw a graph of the cosec function and we have to label all 0, max, min, and asymptotes, and identify the period.*0005

*The key thing to remember here is the cosec(theta) is 1/sin(theta), really this comes down to understanding the graph sin(theta) very well.*0015

*I’m going to start with a graph of sin(theta), that is probably a graph that you should have already memorized.*0029

*My graph of sin(theta) goes up to 1 and down to -1, starts at 0 goes up to 1 and pi/2, down to 0 at pi, down to -1 that is 3pi/2 and back at 0 at 2pi.*0048

*What I have drawn there is not cosec yet, it is sin(theta), in red I will draw cosec(theta).*0073

*Cosec(theta) is just 1/sin(theta), whenever sin is 1 it is 1, whenever sin is 1- it is -1, whenever sin is 0 where you can not divide by 0, that is where cosec has an asymptote.*0083

*It goes up to infinity whenever sin goes down to 0.*0110

*And down to negative infinity whenever sin goes down to 0.*0119

*It looks a lot like the graph of sec did, it got these U’s going up to infinity and upside down U’s going down to negative infinity depending on where the graph of sin hits 0.*0127

*Let us label everything we have been asked to label here, cosec(theta) never crosses the (x) axis, so it has no 0.*0143

*There are no 0 to label there, max and min it has local min at the bottom of each of these U’s.*0158

*This is at pi/2 and 1, that is a local min.*0165

*We have local max at –pi/2 and -1, that is a local max, and another one at 3pi/2, -1, that is our local max.*0174

*We got the max and min, the asymptotes are places where it goes up to infinity and down to negative infinity.*0197

*Let me label those, there is one right there at –pi.*0205

*Here is another one at 0, another one at pi, and finally here is one at 2pi.*0215

*Basically those are the places where sin(theta) is 0, remember cosec is 1/sin(theta) so whenever you are trying to divide by 0, that is where (cosec) blows up to infinity or drops down to negative infinity.*0228

*We got the asymptotes, the period of the (cosec) function is how long it takes to repeat itself.*0242

*That is one period before it starts repeating itself and that is 2pi and that really comes back to the fact that sin(theta) has a period of 2 pi.*0253

*The period of (cosec) just like (sec) is 2 pi.*0262

*It looks like we answered everything there, the key part there is remember that (cosec) is 1/sin, you probably do not need to memorize the graph of (cosec) itself.*0273

*It is good if you are familiar with this general shape of the U’s going up and the U’s going down but you do not need to memorize the details.*0285

*As long as you remember that it is 1/sin, you can work it out from there.*0295

*Ok for our next example here, we are given a bunch of angles and we have to find the (sec) and (cosec) of these common values.*0000

*Let me start by drawing these angles on a unit circle, that is a squish unit circle, let me see if I can do a little better there.*0009

*This is 0, pi/2, pi/, 3pi/2, and 2pi, it looks like at least one of these angles is given in degrees as well.*0036

*I will label the values in degrees, there is 90, 180, 270, and 360, those are the values in degrees.*0049

*Let us figure out where these angles are in the unit circle.*0060

*5pi/6 is over here, 240 degrees is between 180 and 270, that is down here.*0063

*7pi/4 is between 3pi/2 and 2pi, that is down there.*0075

*Remember (sec) and (cosec), those are that reciprocals of (cos) and (sin).*0082

*The smart thing to do here is to figure out the (cos) and (sin) of those angles and then it would be an easy matter to find the (sec) and (cosec).*0087

*I’m going to make a little chart, the angle, (cos), (sin) and then it would be an easy matter to find the (sec) and (cosec).*0099

*Let us start with 5pi/6, that is over here and that is a 30, 60, 90 triangle so we know the values.*0115

*The (cos) is (root 3)/2, (sin) is ½, but then we have to worry about whether they are positive or negative.*0123

*The (x) coordinate is negative so I will make the (cos) negative.*0130

*The (sec) now is just 1/cos, so that is 2/square root 3, but if we rationalized that, that is 2 root 3/3 and of course it is negative.*0135

*The (cosec) is 1/sin so 1/(1/2) is just 2.*0148

*Now the next one is 240 degrees, if you like to translate into radians, that is 4pi/3, that is this angle down here.*0156

*We see a 30, 60, 90 triangle, this time the (cos) is the short side that is ½, the (sin) is the long side root 3/2.*0176

*But it is in the third quadrant so they are both negative.*0187

*The (sec) here is 1/cos, so that is -2, (cosec) 1/sin, 2/root 3 rationalizes to (2 root 3)/3 and that is negative.*0192

*Finally, 7pi/4 that is this angle over here.*0207

*It helps to start out with the (cos) and (sin), that is a 45 degrees triangle so we know they are both root 2/2.*0219

*We just have to figure out which one is positive and which one is negative.*0226

*The (y) coordinate is negative there so the (sin) must be negative.*0230

*The (sec) is 1/(cos), 2 /root 2 is just root 2, (cosec)=1/(sin) is 2/root 2 so that is root 2 again but it is negative.*0235

*I really emphasize to my students that you do not really need to memorize the (sec) and (cosec) of the common values if you really have the (sin) and (cos) memorized well.*0250

*I think it is more important to memorize the (sin) and (cos) very well and the way you get those is by knowing those common values for the 30, 60, 90 triangles and for the 45, 45, 90 triangles.*0264

*You use those to figure out the (sin) and (cos), you figure out which quadrant you are in, which tells you whether they are positive or negative.*0278

*Now you know (sin) and (cos), whether they are positive or negative.*0287

*For (sec) and (cosec), all you have to remember is that sec(theta)=1/cos(theta).*0291

*If you can figure out the (cos), you can figure out (sec).*0300

*Similarly for the cosec(theta) is 1/sin(theta), again if you figure out the sin(theta) it is a simple matter to figure out the cosec(theta).*0303

*I hope all those worked out well for you, this is part of the trigonometry lectures series for www.educator.com.*0315

*Hi welcome back to the trigonometry lectures on educator.com*0000

*Today, we're going to learn about the last trigonometric functions.*0004

*We've learned about the sine and cosine, and tangent and cotangent.*0008

*Today we're going to learn the secant and cosecant function.*0011

*Let's start with their definitions.*0017

*The secant function is just defined by the sec(θ)=1/cos(θ).*0019

*That only works when the cos(θ) is non-zero.*0025

*If the (cosθ)=0, we just say the secant is undefined.*0027

*The cosecant, which people shorten to csc, is just 1/sin(θ).*0033

*If the sin(θ)=0, we just say that the cosecant is undefined.*0040

*Let's start with the first example right away.*0048

*We have to draw a graph of the secant function.*0051

*In particular, we have to label all zeros, max's, mins, and asymptotes and figure what the period of the secant function is.*0055

*Remember now that the secant function, sec(θ), by definition is 1/cos(θ).*0064

*A really good place to start here when you're trying to understand sec(θ) is with the graph of cos(θ).*0073

*Let me start with the graph of cos(θ).*0081

*There's π, there's 2π, and I'm going to extend this out a bit, 3π, and -π.*0086

*Now, remember that the cosine function starts at 1, goes down to 0 at π/2, so there's π/2, goes down to -1 at π, comes back to 0 at 3π/2, back up to 1 at 2π.*0098

*The period of cosine is 2π, so it's repeating itself after 2π.*0130

*I'm not drawing secant yet, I'm drawing cos(x), y=cos(x).*0140

*In black here, I've got, we'll call it cos(θ).*0146

*Now, I'm going to draw the secant function in red.*0153

*That means we're doing 1/cos(x), or 1/cos(θ).*0159

*In particular, 1/1 is 1.*0165

*Then as the cosine goes down to 0, secant is 1/cos, so it goes up to infinity there, and does the same thing on the other side, so secant looks like that.*0170

*When the cosine is negative, when the cosine is -1, secant is -1.*0201

*When the cosine goes to 0, secant blows up but since cosine is negative here, secant goes to negative infinity when the cosine is negative.*0217

*Then when cosine is positive again, secant is positive again, going up to positive infinity on both ends there.*0235

*That's what the secant function looks like.*0247

*While the cosine goes between -1 and 1, secant is just the reciprocal of that, so it goes 1 up to negative infinity, and from -1 down to negative infinity.*0249

*Now, I've got the secant graph in red.*0260

*Let me label all the things that we've been asked to label.*0267

*First of all, zeros, while the sec(θ) has no zeros because it never crosses the x-axis, so there are no zeros to label.*0270

*Max's, all local max's of the secant function, well here's one down here at (π,-1) is a local max.*0282

*Minimum at (0,1) is a local min, (2π,1) is a local min, and so on, as a local min and a local max, every π units.*0295

*We've got the max's and the mins.*0314

*The asymptotes are the places where the secant blows up to infinity or drops down to negative infinity and so that's an asymptote at -π/2, and at π/2.*0318

*We have an asymptote at π/2, and again at 3π/2, and again at 5π/2.*0333

*Basically, every π units, we have an asymptote where the secant function blows up to infinity or drops down to negative infinity.*0350

*Finally, what is the period of secant function.*0362

*The secant function is really dependent on the cosine function, and the cosine function repeats itself once every 2π.*0367

*The period of cosine is 2π.*0374

*The period of secant is also 2π.*0376

*You can see that from the graph, it starts repeating itself after a multiple of 2π.*0384

*That's what the graph of the secant function looks like.*0394

*It kind of has this hues and this upside down hues really dependent on the cosine function because the secant is just 1/cos(θ).*0399

*For the second example, we have to figure out some common values of secant and cosecant at the angles in the first quadrant, 0, π/6, π/4, π/3, and π/2.*0410

*Now, these angles are probably so common that you really should have memorized the sine and cosine.*0426

*I'm going to start by writing down the sine and cosine of these values.*0433

*I'm going to write them down both in degrees and radians because it's very important to be able to identify these common values either way.*0440

*I'll write down degrees, radians, I'll write down the cosine and the sine, and then the secant and cosecant of each one.*0449

*I'll make a nice chart here.*0465

*The values given were 0, π/6, π/4, π/3, and π/2, in terms of degrees, that's 0, 30, 45, 60, and 90.*0467

*Now, you really should have probably memorized the cosine and sine of these already.*0486

*You probably shouldn't even have to check the unit circle.*0493

*But if you need to, go ahead and draw yourself a unit circle.*0496

*Then draw out those triangles in the first quadrant, and you'll be able to figure out the cosine and sine very quickly as long as you remember the values of the 30-60-90 triangles and the 45-45-90 triangles.*0500

*In particular, the cosine and sine of 0, are 1 and 0.*0510

*For 30-degree angle, cosine is root 3 over 2, sine is 1/2.*0517

*For 45-degree angle, they're both root 2 over 2.*0524

*For 60-degree angle, they're just the opposite of what they were for 30, 1/2 and root 3 over 2.*0529

*For 90, they're just the opposite of what they were for 0, 0 and 1.*0537

*Now, the secant is just the reciprocal of cosine, it's just 1/cos.*0542

*I'll just take 1/1, 2 divided by root 3, if you rationalize that, you get 2 root 3 over 3, 2 divided by root 2, is just root 2, the reciprocal of 1/2 is 2, and the reciprocal of 0 is undefined.*0547

*Those are the secants of those common values.*0568

*The cosecant is 1/sin, while 1/0 is undefined, 1/(1/2) is 2.*0574

*The reciprocal of root 2 over 2 is 2 divided by root 2, which again is root 2.*0585

*Then 2 divided by root 3 rationalizes into 2 root 3 over 3, and then 1/1 is just 1.*0592

*I've been really trying to drill you on memorizing the values of sine and cosine.*0601

*I don't think it's really worth memorizing the values of secant and cosecant, they don't come up as often as sine and cosine.*0606

*The key thing to remember is that sec(θ) is just 1/cos(θ), and csc(θ) is just 1/sin(θ).*0615

*As long as you really have memorized the values of sine and cosine, you can always work out the values of secant and cosecant.*0630

*I don't think you need to memorize these values of secant and cosecant.*0639

*It helps if you practice them but it's not really worth memorizing them as long as you know your sine and cosine really well.*0643

*You can always work out secant and cosecant.*0649

*Last thing this example asked is, which other quadrants the secant and cosecant are positive?*0652

*Let's go back and remember our little mnemonic here, All Students Take Calculus.*0662

*That's in quadrant 1, quadrant 2, quadrant 3, and quadrant 4.*0671

*That tells us which of the common functions are positive in which quadrant.*0679

*In the first quadrant, they're all positive, in the second quadrant, only sine, in the third quadrant, only tangent, and in the fourth quadrant, only cosine.*0684

*Let's figure out what that means for secant and cosecant in each case.*0697

*On the first quadrant, they're both positive, both sine and cosine are positive, so secant and cosecant are both positive.*0712

*In the second quadrant, sine is positive, which means that secant is positive, but cosine is negative so secant is negative.*0720

*In the third quadrant, tangent is the only thing that's positive, sine and cosine are both negative, so secant and cosecant are both negative.*0733

*Finally, in the fourth quadrant, cosine is positive so secant is positive, sine is negative so cosecant is negative.*0739

*Now, we have a little chart that tells us which quadrant secant and cosecant are positive and negative in.*0748

*Again, I don't think you really need to memorize this as long as you remember very well where sine and cosine are positive, you can always work out where secant and cosecant are positive and negative.*0756

*For our next example, we're asked to find whether the secant and cosecant functions are odd, even or neither.*0770

*Let's remember what the definition of odd and even are.*0779

*Odd is where f(-x)=-f(x), and that also, by looking at the graph, you can identify odd functions, they have rotational symmetry around the origin.*0784

*Even functions, f(-x)=f(x), and they have mirror symmetry across the y-axis.*0806

*Now, let's look at sec(x), sec(x), actually we have to look at sec(-x) to check whether it's odd or even.*0830

*So, sec(-x), secant remember is 1/cos, 1/cos(-x), cosine is an even function, so this is just 1/cos(x), which is sec(x) again, sec(x) is even.*0840

*Csc(x), well csc(-x), cosecant is 1/sin, so that's 1/sin(-x), but sine's an odd function, so this is 1/-sin(x), which is -csc(x), so cosecant is odd.*0865

*That was just a matter of checking the definitions of odd and even, plugging -x into secant, cosecant, and seeing what we came up with.*0895

*We can also figure it out from the graphs if we remember what those look like.*0904

*Secant, remember ...*0911

*Let me draw a cosine.*0914

*Secant is 1/cos, so that was the one that look like this, that's sec(x) in red there.*0918

*If you look, it has mirror symmetry across the y-axis, which checks that sec(x) is really an even function.*0928

*Mirror symmetry across the y-axis.*0948

*Csc(x), let me draw a quick graph of csc(x).*0955

*Remember, that's based on the graph of sin(x), so start by drawing a graph of sin(x).*0960

*Now, we'll fill in a graph of csc(x), it has asymptotes wherever sine is 0, so that's now our graph of csc(x).*0968

*Clearly, that does not have mirror symmetry across the y-axis, but it does have rotational symmetry around the origin.*0981

*If you spun that around 180 degrees, it would look the same, so it does have rotational symmetry.*0992

*That confirms that cosecant is an odd function.*1005

*We'll try some more examples later.*1017

*You should try working them out yourself, then we'll work them out together.*1019

1 answer

Last reply by: Dr. William Murray

Wed Jul 1, 2015 9:02 AM

Post by Brandon Dorman on June 30, 2015

Hello,

I am trying to wrap my head around this concept. I understand sec=1/cos. but I ran into a problem that has me confused.

The question was:

What is the same as sec(2Ï€/3)?

a. sec(-Ï€/3

b. sec(-5Ï€/3

c. -sec(5Ï€/3)

d. -sec(Ï€/3)

The answer is d. but I'm completely confuse as why.

Thank you!

3 answers

Last reply by: Dr. William Murray

Sat Jul 5, 2014 6:10 PM

Post by Austin Cunningham on June 11, 2014

I have only taken Algebra 1 and Geometry, so I am sometimes utterly confused when you use words I am vaguely familiar with like "asymptotes" and "local minimums" and "local maximums"? Would you mind explaining what exactly the types of max(es) and minimums there are?

1 answer

Last reply by: Dr. William Murray

Tue May 13, 2014 4:46 PM

Post by Kenneth Montfort on May 9, 2014

Could you tell me what lecture stated where sine is an odd function and cosine is an even function?

1 answer

Last reply by: Dr. William Murray

Thu Apr 18, 2013 12:15 PM

Post by abbas esmailzadeh on May 22, 2012

i dont undrstnad if the secant function repeat at 3pi/2 or at2 pi.i am talking about the period.

1 answer

Last reply by: Dr. William Murray

Thu Apr 18, 2013 12:10 PM

Post by Yvonne Hamnquist on February 20, 2012

A tourist wants to determine the height of the Eiffel Tower without looking in her guide book. She observes the angle of the elevation of the top of the tower from one point on the street is 31.73 degrees. She moves 200m closer to the tower and observes an angle of elevation to the top of the tower of 45 degrees. What is the height of the tower to the nearest meter?

Can anyone help?

1 answer

Last reply by: Dr. William Murray

Thu Apr 18, 2013 11:54 AM

Post by David Burns on August 17, 2011

The "Quick Notes" section here does not correspond to the lecture. There are a couple other cases on this course where this happens.