Cosecant, Secant, And Cotangent
Cosecant, Secant, and CotangentIn this chapter we’ll introduced three more trigonometric functions: thecosecant, the secant, and the cotangent. These functions are written as csc(θ),sec(θ), and cot(θ) respectively. They are the functions defined by the formulasbelow:csc(θ) 1sin(θ)sec(θ) 1cos(θ)cot(θ) cos(θ)sin(θ)Graphs of cosecant, secant, and cotangent00274
PeriodsCosecant, secant, and cotangent are periodic functions. Cosecant and secant have the same period as sine and cosine do, namely 2π. Cotangenthas period π, just as tangent does. In terms of formulas, the previous twosentences mean thatcsc(θ 2π) csc(θ)sec(θ 2π) sec(θ)cot(θ π) cot(θ)It’s easy to check why these functions have the periods that they do. Forexample, because sine has period 2π—that is, because sin(θ 2π) sin(θ)—we can check that11 csc(θ)csc(θ 2π) sin(θ 2π) sin(θ)Similarly, the secant function has the same period, 2π, as the function usedto define it, cosine.Even and oddRecall that an even function is a function f (x) with the property thatf ( x) f (x). Examples include x2 , x4 , x6 , and cosine.We can add secant to the list of functions that we know are even functions.That is, sec( θ) sec(θ). The reason secant is even is that cosine is even:sec( θ) 11 sec(θ)cos( θ) cos(θ)An odd function is a function f (x) with the property that f ( x) f (x).Examples include x3 , x5 , x7 , sine, and tangent.Cosecant and cotangent are odd functions, meaning that csc( θ) csc(θ)and cot( θ) cot(θ). We can check that these identities are true by usingthat sine is an odd function and that cosine is even:csc( θ) 11 csc(θ)sin( θ) sin(θ)cot( θ) cos( θ)cos(θ) cot(θ)sin( θ) sin(θ)275
Cofunction identitiesSine and cosine, secant and cosecant, tangent and cotangent; these pairs offunctions satisfy a common identity that is sometimes called the cofunctionidentity:π2 θ cos(θ) sec π2 θ csc(θ) tan π2 θ cot(θ)sinThese identities also “go the other way”:π2 θ sin(θ) csc π2 θ sec(θ) cot π2 θ tan(θ)cos Let’s check one of these six identities, the identity cos π2 θ sin(θ). Inorder to see that this identity is true, we’ll start with cos π2 θ and we’lluse that cosine is an even function, so π hπi π cos θ cos θ cos θ 222 πNow we can use the identity cos θ 2 sin(θ) (which is Lemma 9 fromthe Sine and Cosine chapter) so that we have ππ θ cos θ cos sin(θ)22as we had claimed. Using the cofunction identity that we just examined, sin(θ) cos π2 θ ,we can check that the first cofunction identity from the list above is true: π π hπi sin θ cos θ cos(θ)222276
ExercisesFor #1-12, use the chart on page 227 in the chapter “Sine and Cosine” and11that csc(θ) sin(θ), sec(θ) cos(θ), and cot(θ) cos(θ)sin(θ) to find the given value. 1.) csc π6 2.) csc π4 3.) csc π3 4.) csc π25.) sec(0) 6.) sec π6 7.) sec π4 8.) sec π39.) cotπ6 10.) cotπ4 11.) cotπ3 12.) cotπ2 Find the solutions of the following equations in one variable.13.) loge (x) loge (12) loge (x 1)14.) (x 4)2 3615.) e3x 2 4277
Match the numbered piecewise defined functions with their lettered graphsbelow.π2orπ2 x π(csc(x)17.) g(x) 2if 0 x if x π2π2orπ2 x π(cot(x)18.) h(x) 2if 0 x if x π2π2orπ2 x π(cot(x)19.) p(x) 2if 0 x if x π2π2orπ2 x πIif 0 x if x π2I(csc(x)16.) f (x) 2B.)C.)D.)IIIIIIA.)278
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( )
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/
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
of cantilevered, braced, and/or anchored secant and tangent pile retaining wall systems. Secant and/or tangent pile walls may be constructed using various technologies including . A reinforced concrete template that is formed in the ground prior to the start of the construction of the secant or tangent pile walls. The guidewall provides .
Introduction to Literary Criticism. Definition and Use “Literary criticism” is the name given to works written by experts who critique—analyze—an author’s work. It does NOT mean “to criticize” as in complain or disapprove. Literary criticism is often referred to as a “secondary source”. Literary Criticism and Theory Any piece of text can be read with a number of different .
