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 ).
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.
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.
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 ).