Cosecant, Secant & Cotangent
Cosecant, Secant &Cotangentmc-TY-cosecseccot-2009-1In this unit we explain what is meant by the three trigonometric ratios cosecant, secant andcotangent. We see how they can appear in trigonometric identities and in the solution oftrigonometrical equations. Finally, we obtain graphs of the functions cosec θ, sec θ and cot θfrom knowledge of the related functions sin θ, cos θ and tan θ.In order to master the techniques explained here it is vital that you undertake the practiceexercises provided.After reading this text, and/or viewing the video tutorial on this topic, you should be able to: define the ratios cosecant, secant and cotangent plot graphs of cosec θ, sec θ and cot θContents1. Introduction22. Definitions of cosecant, secant and cotangent23. The graph of cosec θ44. The graph of sec θ55. The graph of cot θ6www.mathcentre.ac.uk1c mathcentre 2009
1. IntroductionThis unit looks at three new trigonometric functions cosecant (cosec), secant (sec) and cotangent (cot). These are not entirely new because they are derived from the three functions sine,cosine and tangent.2. Definitions of cosecant, secant and cotangentThese functions are defined as follows:Key Pointcosec θ 1sin θsec θ 1cos θcot θ 1tan θThese functions are useful in the solution of trigonometrical equations, they can appear in trigonometric identities, and they can arise in calculus problems, particularly in integration.ExampleConsider the trigonometric identitysin2 θ cos2 θ 1Suppose we divide everything on both sides by cos2 θ. Doing this producessin2 θ cos2 θ1 22cos θ cos θcos2 θThis can be rewritten as sin θcos θ 2 1 1cos θ 2that is astan2 θ 1 sec2 θThis, in case you are not already aware, is a common trigonometrical identity involving sec θ.ExampleConsider again the trigonometric identitysin2 θ cos2 θ 1Suppose this time we divide everything on both sides by sin2 θ; this producessin2 θ cos2 θ1 22sin θ sin θsin2 θwww.mathcentre.ac.uk2c mathcentre 2009
This can be rewritten as1 cos θsin θ 2 1sin θ 2that is as1 cot2 θ cosec2 θAgain, we see one of our new trigonometric functions, cosec θ, appearing in an identity.ExampleSuppose we wish to solve the trigonometrical equationcot2 θ 3for 0 θ 360 We begin the solution by taking the square root: cot θ 3orIt then follows thatInverting we find 3 1 3tan θor 1tan θ 3or1 331The angle whose tangent is is one of the special angles described in the unit Trigonometrical31ratios in a right-angled triangle. In fact is the tangent of 30 . So this is one solution of the3equation tan θ 13 . What about other solutions ?We refer to a graph of the function tan θ as shown in Figure 1.tan θ 13- 13210o30oo90o180oo270360θo150o330Figure 1. A graph of tan θ.From the graph we see that the next solution of tan θ 13is 210 (that is 180 further along).From the same graph we can also deduce, by consideration of symmetry, that the angles whose1tangent is are 150 and 330 .3www.mathcentre.ac.uk3c mathcentre 2009
In summary, the equation cot2 θ 3 has solutionsθ 30 , 150 , 210 , 330 So, solving equations involving cosec, sec and cot can often be solved by simply turning theminto equations involving the more familiar functions sin, cos and tan.3. The graph of cosec θWe study the graph of cosec θ by first studying the graph of the closely related function sin θ,one cycle of the graph of which is shown in Figure 2.sin θA1DC090 o180o-1270 o360oθBFigure 2. A graph of sin θ.The graph of cosec θ can be deduced from the graph of sin θ because cosec θ sin1 θ . Note thatwhen θ 90 , sin θ 1 and hence cosec θ 1 as well. Similarly when θ 270 , sin θ 1 andhence cosec θ 1 as well. These observations enable us to plot two points on the graph ofcosec θ. The corresponding points are marked A and B in both Figures 2 and 3. When θ 0,sin θ 0, but because we can never divide by 0 we cannot evaluate cosec θ in this way. However,note that if θ is very small and positive (i.e. close to, but not equal to zero) sin θ will be small1and positive, and hencewill be large and positive. Points marked C on the graphs representsin θ this. Similarly when θ 180 , sin θ 0 and again we cannot divide by zero to find cosec 180 .Suppose we look at values of θ just below 180 . Here, sin θ is small and positive, so once againcosec θ will be large and positive (points D).These observations enable us to gradually build up the graph as shown in Figure 3. The verticalwww.mathcentre.ac.uk4c mathcentre 2009
dotted lines on the graph are called asymptotes.cosec θC1DA090o-1180 o270 o360oθBFigure 3. A graph of cosec θ.Note that when θ is just slightly greater than 180 then sin θ is small and negative, so that cosec θis large and negative as shown in Figure 3. Continuing in this way the full graph of cosec θ canbe constructed.In Figure 2 we showed just one cycle of the sine graph. This generated one cycle of the graphof cosec θ. Clearly, if further cycles of the sine graph are drawn these will generate further cyclesof the cosecant graph. We conclude that the graph of cosec θ is periodic with period 2π.4. The graph of sec θWe can draw the graph of sec θ by first studying the graph of the related function cos θ one cycleof which is shown in Figure 4.cos θ10-1AC90o D180 o270o360oθBFigure 4. A graph of cos θ.Note that when θ 0, cos θ 1 and so sec 0 1. This gives us a point (A) on the graph.Similarly when θ 180 , cos θ 1 and so sec 180 1 (Point B). When θ 90 , cos θ 0and so we cannot evaluate sec 90 . We proceed as before and look a little to the left and right.When cos θ is small and positive, cos1 θ will be large and positive. This gives point C. When cos θis small and negative, cos1 θ will be large and negative. This gives point D. Continuing in this waywe can produce the graph shown in Figure 5.www.mathcentre.ac.uk5c mathcentre 2009
Recall that we have only shown one cycle of the cosine graph in Figure 4. However because thisrepeats with a period of 2π it follows that the graph of sec θ is also periodic with period 2π.sec θCA10-190 o180oθ360o270oBDFigure 5. A graph of sec θ.5. The graph of cot θWe can draw the graph of cot θ by first studying the graph of tan θ two cycles of which are shownin Figure 6.BtanθA090 180 270360 θFigure 6. A graph of tan θ.We proceed as before. When θ is small and positive (just above zero), so too is tan θ. So cot θwill be large and positive (point A). When θ is close to 90 the value of tan θ is very large andpositive, and so cot θ will be very small (point B). In this way we can obtain the graph shown inwww.mathcentre.ac.uk6c mathcentre 2009
Figure 7. Because the tangent graph is periodic with period π, so too is the graph of cot θ.Acot θB090 o180o270 o360 oθFigure 7. A graph of cot θ.In summary, we have now met the three new trigonometric functions cosec, sec and cot andobtained their graphs from knowledge of the related functions sin, cos and tan.Exercises1. Use the values of the trigonometric rations of the special angles 30o , 45o and 60o to determine the following without using a calculatora) cot 45od) cosec 2 45og) cot 315ob)e)h)cosec 30oc)2ocot 60f)ocosec ( 30 ) i)sec 60osec2 30osec 240o2. Find all the solutions of each of the following equations in the range stated (give youranswers to 1 decimal place)(a) cot θ 0.2 with 0o θ 360o(b) cosec θ 4 with 0o θ 180o(c) cosec θ 4 with 0o θ 360o(d) cosec θ 4 with 180o θ 180o(e) sec θ 4 with 0o θ 180o(f) sec θ 4 with 0o θ 360o(g) sec θ 4 with 180o θ 180o(h) cot θ 0.5 with 0o θ 360o(i) cosec θ 0.5 with 0o θ 360o(j) sec θ 0.5 with 0o θ 360o3. Determine whether each of the following statements is true or false(a) cot θ is periodic with period 180o .(b) cosec θ is periodic with period 180o .www.mathcentre.ac.uk7c mathcentre 2009
(c) Since the graph of cos θ is continuous, the graph of sec θ is continuous.(d) cosec θ never takes a value less than 1 in magnitude.(e) cot θ takes all values,Answers1. a) 1b) 2c) 2d) 2e)13f)43g) -1 h) -2 i) -22. a) 78.7o , 258.7o b) 14.5 o , 165.5 o c) 14.5 o , 165.5 o d) 14.5 o , 165.5 o e) 75.5of) 75.5o , 284.5o g) 75.5o , -75.5o h) 63.4o , 243.4 o i) No solutions j) No solutions3. a) True b) False c) False d) True e) Truewww.mathcentre.ac.uk8c mathcentre 2009
Cosecant, Secant & Cotangent mc-TY-cosecseccot-2009-1 In this unit we explain what is meant by the three trigonometric ratios cosecant, secant and cotangent. We see how they can appear in trigonometric identities and in the solution of trigonometrical equations. Finally, we obtain graphs of the functions cosecθ, secθ and cotθ
Graphing & Writing Secant, Cosecant, Tangent and Cotangent Functions Date Topic Assignment Did It Fri 11/2 4.6 Graphing Secant and Cosecant NOTES p.1 & 2 Worksheet p. 3, 4 (#5 – 12) Mon 11/5 4.6 Graphing Tangent and Cotangent NOTES p. 5 Worksheet p. 6, 7 (#3 – 10) Tues 11/
Cosecant, secant, and cotangent are periodic functions. Cosecant and se-cant have the same period as sine and cosine do, namely 2ˇ. Cotangent has period ˇ, just as tangent does. In terms of formulas, the previous two sentences mean that csc( 2ˇ) csc( ) sec( 2ˇ) sec( ) cot( ˇ) cot( )
becomes corresponding to secant i.e. cosecant. Cosecant with angle Ө secant (hypotenuse) / Opposite side (cotangent) cosec Ө AC / AB The defination of Cosecant : - it is the ratio of hypotenus i.e.secant to the opposite side of given triangle. The reason for main function sine , secant , tangent :- Sine is only use for chord which is .
Section 7.3 The Graphs of the Tangent, Cosecant, Secant, and Cotangent Functions. OBJECTIVE 1: Understanding the Graph of the Tangent Function and its Properties Recall that y tanx can equivalently be written y sin x cosx. Recall also that division by zero is never allowed. Therefore, the domain of
Precalculus: Graphs of Tangent, Cotangent, Secant, and Cosecant The Cotangent Function The tangent function is cotx cosx sinx. It will have zeros where the cosine function has zeros, and vertical asymptotes where the sine function has zeros. It will look like the cosine function where the sine is essentially equal to 1, which is when xis near .
Figure 3 shows a right triangle with two legs: cosecant and secant, and the hypotenuse equals to the product of secant*cosecant. If we split those two legs and hypotenuse into parts that obey the condition of the Divided Line of Plato then we might get the expressions for the sensible energy, latent energy, sensible momentum, latent
The line containing P and Q is a secant line. (Refer to Section 1.3.) The slope m sec of the secant line is Now look at Figure 5. As we move along the graph off from Q toward P, we obtain a succession of secant lines. The closer we get to P, the closer the secant line is to the tangent line. The limit-ing p
N. Sharma, M. P. Rai* Amity Institute of Biotechnology (J-3 block), Amity University Uttar Pradesh, Sector-125, Noida, India A B S T R A C T P A P E R I N F O Media requirement for microalgae cultivation adds most to the cost of biodiesel production at Paper history: Received 30 July 2015 Accepted in revised form 6 September 2015 Keywords: - Chlorella pyrenoidosa Cattle waste Lipid content .
