sec inverse 0 and cosec inverse 0
Answers
Answer:
Lastly we apply the same methods used for inverse sine and cosine to construct inverses for secant and cosecant.
Inverse Secant and Cosecant
Defining sec -1(x) and csc -1(x)
Using the procedures above we arrive at definitions for these two inverse trigonometric functions.
For x in (- , -1] or [1 , ), sec -1(x) is the angle measure in [0 , /2) or (/2 , ] whose secant value is x.
For x in (- , -1] or [1 , ), csc -1(x) is the angle measure in [-/2 , 0) or (0 , /2] whose cosecant value is x.
Inverse Properties
We have the usual composition formulas.
sec -1(sec(x)) = x for x in [0 , /2) or (/2 , ].
sec(sec -1(x)) = x for x in (- , -1] or [1 , ).
csc -1(csc(x)) = x for x in [-/2 , 0) or (0 , /2].
csc(csc -1(x)) = x for x in (- , -1] or [1 , ).
Because of the intervals chosen we get this identity, similar to the one stated for inverse sine and cosine.
csc -1(x) = /2 - sec -1(x) for x in (- , -1] or [1 , ).
To get used to thinking inversely, try this exercise without a calculator. The answers involve familiar angles and can be found just using a sketch of the angles or graphs.
Inverses and Familiar Angles
Using a calculator
It is also important to be proficient at using your calculator to find inverse values. Use radian mode and round answers to four decimal places in the next exercise. Since most calculators don't have "sec-1" or "csc-1" buttons, you will have to use the fact that if csc(A) = x then sin(A) = 1/x or the fact that if sec(A) = x then cos(A) = 1/x.
Using a Calculator
Other notation for sec-1 include arcsec and asec. Similarly, arccsc and acsc are used for csc-1. Typically, we use asec and acsc in the interactive exercises and demonstrations because of ease of typing and for computer recognition of the function.