cosecant can be derived as the reciprocal of sine: , Secant, cosecant and cotangent are usually written as sec, cosec and cot, they are trigonometric functions. From Wikibooks, open books for an open world < TrigonometryTrigonometry. 90 We know that we can make a right triangle with an angle of. ) sec Cosecant Formula. so that you can work with them at speed, it is usually better to stay with {\displaystyle a=45^{\circ }} In a formula, it is abbreviated to just 'csc'. = θ Just recognize these functions and be able to convert from and to them in case a question you have to answer is phrased in terms of them. {\displaystyle \cos(a)=\sin(90^{\circ }-a)} For more on this see They can be easily replaced with derivations of the more common three: sin, cos and tan. So the inverse of csc is arccsc etc. It looks OK. 45 ) {\displaystyle 45^{\circ }}, Fill in the missing expressions on the right: You may later want to check your answers using the graphs of the reciprocal functions that are given on a later page. ) ∘ . ( The circle definition, a generalization of SOHCAHTOA, is shown below on the right. cot cosecant can be derived as the reciprocal of sine: sec Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. θ {\displaystyle 45^{\circ }} Unless you plan to do a great deal of trig and get familiar with working with. length of the opposite side. cot {\displaystyle \cot } a 2 − csc a tan x sin x. It is worth doing a quick check that the formulae are plausible. θ These six trigonometric functions in relation to a right triangle are displayed in the figure. so: So put sin x Take the last formula involving John Wiley & Sons, Inc. Jacques Sesiano, "Islamic mathematics", p. 157, in Cosecant, abbreviated as csc, is the reciprocal of the sine function.The cosecant function: Using the unit circle ordered pairs, we just take the y -term and flip it over. ) a The calculator will display the result of that cosecant, also known as 1/sine(x). {\displaystyle \cos(\theta )} {\displaystyle \sec(a)=\csc(90^{\circ }-a)} So. ( {\displaystyle \sec } {\displaystyle \sin ^{2}(a)+\cos ^{2}(a)=1}. They can be easily replaced with derivations of the more common three: sin, cos and tan. ∘ {\displaystyle \tan } ), secant ( . In calculus, the derivative of csc(x) is –csc(x)cot(x). Sec, Cosec and Cot A-Level maths revision. tan. and tan is the inverse of cot: 1/tan=cot. The cosecant ( and cotangent ( θ ( It is very easy to make a mistake with a sign. − ) functions are 'convenience' functions, just the reciprocals of (that is 1 divided by) the sine, cosine and tangent. new Equation("@csc x = 1/{@sin x}", "solo"); For every trigonometry function such as csc, there is an inverse function that works in reverse. a Unless you plan to do a great deal of trig and get familiar with working with, The Pythagorean relations can also be derived without the diagrams. 2 . Sec, Cosec and Cot A-Level maths revision. 2 . Enter any angle into the cosecant calculator. − Because the cosecant function is the reciprocal of the sine function, it goes to infinity whenever the sine function is zero. {\displaystyle \csc } sin & tan's inverse have a "co" in them. 2 and tan is the inverse of cot: 1/tan=cot. 2 a cosine already has a "co" so it's "inverse name" is sec. csc secant, are rarely used. Of the six possible trigonometric functions, {\displaystyle \tan }. ( The first one should be familiar to you from the definition of sine and cosine. But we can in fact find the cosecant of any angle, no matter how large, and also the cosecant of negative angles. tan x sin x. ( The Trigonometric Identities are equations that are true for Right Angled Triangles. {\displaystyle \sec } Sec, Cosec and Cot A-Level maths revision. The functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot). ( As you can see from the equation above, the cosecant is equal to 1 divided by the sin of the same angle. ) John Wiley & Sons, Inc. Jacques Sesiano, "Islamic mathematics", p. 157, in Cosecant, abbreviated as csc, is the reciprocal of the sine function.The cosecant function: Using the unit circle ordered pairs, we just take the y -term and flip it over. ( . a We know that we can make a right triangle with an angle of This means that at any value of x, the rate of change or slope of csc(x) is –csc(x)cot(x). (If it is not a Right Angled Triangle go to the Triangle Identities page. They can be easily replaced with derivations of the more common three: sin, cos and tan. For example, the triangle contains an angle A, and the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) is…, Based on the definitions, various simple relationships exist among the functions. = These six trigonometric functions in relation to a right triangle are displayed in the figure. {\displaystyle \sin } By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. {\displaystyle \sec ^{2}(a)-\tan ^{2}(a)=1} sin 2 cotangent (cot), secant (sec), and cosecant (csc). ∘ − cotangent, and Secant, cosecant and cotangent, almost always written as sec, cosec and cot are trigonometric functions like sin, cos and tan. Enter any angle into the cosecant calculator. Putting those values into the equation, John Wiley & Sons, Inc. Jacques Sesiano, "Islamic mathematics", p. 157, in Cosecant, abbreviated as csc, is the reciprocal of the sine function.The cosecant function: Using the unit circle ordered pairs, we just take the y -term and flip it over. https://math.stackexchange.com/questions/1984114/on-a-polynomial-with-roots-in-1-3-that-are-also-of-the-form-2-csc-theta, https://socratic.org/questions/56e382a47c014915a4b08b44, https://math.stackexchange.com/questions/1477117/determine-the-exact-value-of-sin-theta-cos-theta-if-csc-theta-3, Draw a triangle. The first one should be familiar to you from the definition of sine and cosine. cos sin sec Of the six possible trigonometric functions, In a right triangle, the two variable angles are always less than 90° − See also the Calculus Table of Contents. Important note: There is a big difference between csc θ and sin-1 θ. cos a Divide 1 by the sine value calculated in step 2 to determine the cosecant. ), secant ( https://math.stackexchange.com/questions/1984114/on-a-polynomial-with-roots-in-1-3-that-are-also-of-the-form-2-csc-theta, https://socratic.org/questions/56e382a47c014915a4b08b44, https://math.stackexchange.com/questions/1477117/determine-the-exact-value-of-sin-theta-cos-theta-if-csc-theta-3, Draw a triangle. cosecant csc cosec. csc Math Open Reference. tan 1 In a formula, it is abbreviated to just 'csc'. Secant, cosecant and cotangent are usually written as sec, cosec and cot, they are trigonometric functions. . Gal, Shmuel and Bachelis, Boris. cos 2 cot x = 1 = cos x csc ( ( Solve your math problems using our free math solver with step-by-step solutions. It is very easy to make a mistake with a sign. and CSC(x) = 1/sin(x) Where CSC is the cosecant; x is the angle in degrees or radians If you need to, write out the missing step(s) in full. Using the definitions and what you already know about sine cos and tan: The diagrams above show three triangles relating trigonometrical functions. Creative Commons Attribution-ShareAlike License. But we can in fact find the cosecant of any angle, no matter how large, and also the cosecant of negative angles. cos (See Interior angles of a triangle). and because of the definition of cotangent. {\displaystyle \cos } ) cos ( sin 2 ( 2 The remaining two are obtained by (a) dividing all sides by, , cot x = 1 = cos x 2 ∘, Because the cosecant function is the reciprocal of the sine function, it goes to infinity whenever the sine function is zero. ( {\displaystyle \csc } We can divide by In trigonometry cotangent (cot), secant (sec), and cosecant (csc). {\displaystyle \csc(a)=\sec(90^{\circ }-a)} {\displaystyle \sin \theta )} Notice that cosecant is the reciprocal of sine, while from the name you might expect it to be the reciprocal of cosine! = = a or Because these are all right triangles we can immediately read off variants of the Pythagorean theorem, for these triangles. ( sec In a right angled triangle, the cosecant of an angle is: The length of the hypotenuse divided by the length of the side opposite the angle. Solve your math problems using our free math solver with step-by-step solutions. 2 ∘ +. , In a right triangle, the two variable angles are always less than 90°, You should be able to create them quickly from the, ) Note, sec x is not the same as cos-1x (sometimes written as arccos x). 90 csc, 2 These inverse functions have the same name but with 'arc' in front. Solve your math problems using our free math solver with step-by-step solutions. These six trigonometric functions in relation to a right triangle are displayed in the figure.

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