inversion of trigonometry funtions
Answers
Answer:
You've studied how the trigonometric functions sin(x) , cos(x) , and tan(x) can be used to find an unknown side length of a right triangle, if one side length and an angle measure are known. The inverse trigonometric functions sin−1(x) , cos−1(x) , and tan−1(x) , are used to find the unknown measure of an angle of a right triangle when two side lengths are known.
Graphs of Inverse Trigonometric Functions
Trigonometric functions are all periodic functions . Thus the graphs of none of them pass the Horizontal Line Test and so are not 1−to−1 . This means none of them have an inverse unless the domain of each is restricted to make each of them 1−to−1 . Since the graphs are periodic, if we pick an appropriate domain we can use all values of the range .
If we restrict the domain of f(x)=sin(x) to [−π2,π2] we have made the function 1−to−1 . The range is [−1,1] .
(Although there are many ways to restrict the domain to obtain a 1−to−1 function this is the agreed upon interval used.)
We denote the inverse function as y=sin−1(x) . It is read y is the inverse of sine x and means y is the real number angle whose sine value is x . Be careful of the notation used. The superscript “ −1 ” is NOT an exponent. To avoid this notation, some books use the notation y=arcsin(x) instead.
To graph the inverse of the sine function, remember the graph is a reflection over the line y=x of the sine function.
Notice that the domain is now the range and the range is now the domain. Because the domain is restricted all positive values will yield a 1st quadrant angle and all negative values will yield a 4th quadrant angle.
Similarly, we can restrict the domains of the cosine and tangent functions to make them 1−to−1 .
The domain of the inverse cosine function is [−1,1] and the range is [0,π] . That means a positive value will yield a 1st quadrant angle and a negative value will yield a 2nd quadrant angle.
The domain of the inverse tangent function is (−∞,∞) and the range is (−π2,π2) . The inverse of the tangent function will yield values in the 1st and 4th quadrants.
The same process is used to find the inverse functions for the remaining trigonometric functions--cotangent, secant and cosecant.
Answer:
The inverse trigonometric functions sin−1(x) , cos−1(x) , and tan−1(x) , are used to find the unknown measure of an angle of a right triangle when two side lengths are known.
Step-by-step explanation: