Solving a linear equation involving the sine function
Find all possible exact solutions for the equation
Solving for all possible values of
t means that solutions include angles beyond the period of
From
[link] , we can see that the solutions are
and
But the problem is asking for all possible values that solve the equation. Therefore, the answer is
Solving equations involving a single trigonometric function
When we are given equations that involve only one of the six trigonometric functions, their solutions involve using algebraic techniques and the unit circle (see
[link] ). We need to make several considerations when the equation involves trigonometric functions other than sine and cosine. Problems involving the reciprocals of the primary trigonometric functions need to be viewed from an algebraic perspective. In other words, we will write the reciprocal function, and solve for the angles using the function. Also, an equation involving the tangent function is slightly different from one containing a sine or cosine function. First, as we know, the period of tangent is
not
Further, the domain of tangent is all real numbers with the exception of odd integer multiples of
unless, of course, a problem places its own restrictions on the domain.
Solving a problem involving a single trigonometric function
Solve the problem exactly:
As this problem is not easily factored, we will solve using the square root property. First, we use algebra to isolate
Then we will find the angles.
Recall that the tangent function has a period of
On the interval
and at the angle of
the tangent has a value of 1. However, the angle we want is
Thus, if
then
