Video Transcript
Find the set of values satisfying cos of three 𝑥 plus 𝜋 over two equals one-half, where 𝑥 is greater than or equal to zero and less than two 𝜋.
In this question, we are asked to find all the solutions to the trigonometric equation that are greater than or equal to zero and less than two 𝜋 radians. We will begin by letting the argument three 𝑥 plus 𝜋 over two equal 𝜃. And as a result, we need to solve the equation cos 𝜃 equals one-half. Multiplying our inequality by three, we see that three 𝑥 must be greater than or equal to zero and less than six 𝜋. We can then add 𝜋 over two to each part of the inequality such that three 𝑥 plus 𝜋 over two is greater than or equal to 𝜋 over two and less than 13𝜋 over two. And in turn 𝜃 lies between these two values.
Next, we will sketch the graph of 𝑦 equals cos 𝜃 between 𝜋 over two and 13𝜋 over two. And we need to find the values of 𝜃 for which cos 𝜃 is equal to one-half. The horizontal line 𝑦 is equal to one-half intersects the curve 𝑦 equals cos 𝜃 six times between 𝜋 over two and 13𝜋 over two. And we can calculate these values of 𝜃 using the symmetry of the curve.
Recalling our special angles, we know that the cos of 𝜋 over three radians or 60 degrees is equal to one-half. This means that our first two points of intersection have values of 𝜃 equal to two 𝜋 minus 𝜋 over three and two 𝜋 plus 𝜋 over three. The next two solutions are 𝜃 equals four 𝜋 minus 𝜋 over three and four 𝜋 plus 𝜋 over three.
Finally, using the symmetry of the cosine curve once again, we have 𝜃 is equal to six 𝜋 minus 𝜋 over three and six 𝜋 plus 𝜋 over three. Simplifying these six values, we have 𝜃 is equal to five 𝜋 over three, seven 𝜋 over three, 11𝜋 over three, 13𝜋 over three, 17𝜋 over three, and 19𝜋 over three. Returning to our original substitution, this means that three 𝑥 plus 𝜋 over two is equal to these six values.
We can then subtract 𝜋 over two from each of our solutions such that three 𝑥 is equal to seven 𝜋 over six, 11𝜋 over six, 19𝜋 over six, 23𝜋 over six, 31𝜋 over six, and 35𝜋 over six. Finally, dividing through by three, we obtain the following six solutions. The set of values that satisfy the equation cos of three 𝑥 plus 𝜋 over two equals one-half, where 𝑥 is greater than or equal to zero and less than two 𝜋, are seven 𝜋 over 18, 11𝜋 over 18, 19𝜋 over 18, 23𝜋 over 18, 31𝜋 over 18, and 35𝜋 over 18.
