24 Nisan 2011 Pazar

Part 3: Inverse Trigonometric Functions And Equations

Inverse Cosine and Inverse Sine

The standard trig functions are periodic, meaning that they repeat themselves. Therefore, the same output value appears for multiple input values of the function. This makes inverse functions impossible to construct. In order to solve equations involving trig functions, it is imperative for inverse functions to exist. Thus, mathematicians have to restrict the trig function in order create these inverses.

To define an inverse function, the original function must be one-to-one. For a one-to-one correspondence to exist, (1) each value in the domain must correspond to exactly one value in the range, and (2) each value in the range must correspond to exactly one value in the domain. The first restriction is shared by all functions; the second is not. The sine function, for example, does not satisfy the second restriction, since the same value in the range corresponds to many values in the domain (see Figure 1 ).





Figure 1
Sine function is not one to one.





Figure 2
Graph of restricted cosine function.

The inverse cosine function is defined as the inverse of the restricted Cosine function Cos−1(cos x) = x≤ x ≤ π. Therefore,




Figure 3
Graph of inverse cosine function.





Identities for the cosine and inverse cosine:




The inverse sine function's development is similar to that of the cosine. The restriction that is placed on the domain values of the sine function is




This restricted function is called Sine (see Figure 4 ). Note the capital “S” in Sine.





Figure 4
Graph of restricted sine function.

The inverse sine function (see Figure 5 ) is defined as the inverse of the restricted Sine function y = Sin x,





Figure 5
Graph of inverse sine function.





Therefore,




Identities for the sine and inverse sine:








Figure 6
Symmetry of inverse sine and cosine.

Example 1: Using Figure 7 , find the exact value of Cos−1 .





Figure 7
Drawing for Example 1.





Thus, y = 5π/6 or y = 150°.
Example 2: Using Figure 8 , find the exact value of Sin−1 .





Figure 8
Drawing for Example 2.





Thus, y = π/4 or y = 45°.
Example 3: Find the exact value of cos (Cos−1 0.62).
Use the cosine-inverse cosine identity:



Other Inverse Trigonometric Functions

To define the inverse tangent, the domain of the tangent must be restricted to





This restricted function is called Tangent (see Figure 1 ). Note the capital “T” in Tangent.





Figure 1
Graph of restricted tangent function.

The inverse tangent function (see Figure 2 ) is defined as the inverse of the restricted Tangent function y = Tan x,





Figure 2
Graph of inverse tangent function.





Therefore,




Identities for the tangent and inverse tangent:








Figure 3
Graphs of inverse cotangent, inverse secant, and inverse cosecant functions.

Trigonometric identities involving inverse cotangent, inverse secant, and inverse cosecant:




Example 1: Determine the exact value of sin [Sec−1 (−4)] without using a calculator or tables of trigonometric functions.








Figure 4
Drawing for Example 1.





Therefore,




Example 2: Determine the exact value of cos (Tan−1 7) without using a calculator or tables of trigonometric functions.








Figure 5
Drawing for Example 2.





Therefore,




Trigonometric Equations

Trigonometric identities are true for all replacement values for the variables for which both sides of the equation are defined. Conditional trigonometric equations are true for only some replacement values. Solutions in a specific interval, such as 0 ≤ x ≤ 2π, are usually calledprimary solutions. A general solution is a formula that names all possible solutions.

The process of solving general trigonometric equations is not a clear-cut one. No rules exist that will always lead to a solution. The procedure usually involves the use of identities, algebraic manipulation, and trial and error. The following guidelines can help lead to a solution.
If the equation contains more than one trigonometric function, use identities and algebraic manipulation (such as factoring) to rewrite the equation in terms of only one trigonometric function. Look for expressions that are in quadratic form and solve by factoring. Not all equations have solutions, but those that do usually can be solved using appropriate identities and algebraic manipulation. Look for patterns. There is no substitute for experience.
Example 1: Find the exact solution:




First, transform the equation by using the identity sin2 α + cos2α = 1.




Therefore,




Thus,




Example 2: Solve cos 2 x = 3(sin x − 1) for all real values of x.




The first answer, −2.351, is not a solution, since the sine function must range between − 1 and 1. The second answer, 0.8508, is a valid value. Thus, if k is an integer,




In radian form,




In degree form,




Example 3: Find the exact solution:




First, transform the equation by using the double angle identity cos 2θ = 2 cos2θ − 1.




Therefore,




Thus,




The Expression M sin Bt + N cos Bt

The equation y = M sin Bt + N cos Bt and the equation y = A sin (Bt + C) are equivalent where the relationships of A, B, C, M, and N are as follows. The proof is direct and follows from the sum identity for sine. The following is a summary of the properties of this relationship.

M sin Bt + N cos Bt =  sin (Bt + C) given that Cis an angle with a point P(M, N) on its terminal side (see Figure 1 ). 






Figure 1
Reference graph for y = M sin Bt + N cos Bt.