Find the set of values satisfying root two sin 𝜃 cos 𝜃 minus cos 𝜃 equals zero, where 𝜃 is greater than or equal to zero degrees but less than 360 degrees.
We notice from the equation that cos 𝜃 is a common term. This means that we can factorise out cos 𝜃. Factorising gives us cos 𝜃 multiplied by root two sin 𝜃 minus one. This means that we have two solutions. Either cos 𝜃 is equal to zero or root two sin 𝜃 minus one is equal to zero.
Adding one to both sides of the second equation gives us root two sin 𝜃 is equal to one. Dividing both sides of this equation by root two gives us sin 𝜃 is equal to one over root two.
We now need to find all of the solutions for both of these equations. The cosine or cos graph is shown in the diagram. There’re two values where this is equal to zero. 𝜃 equals 90 degrees. And 𝜃 equals 270 degrees. We therefore have two solutions for the cos 𝜃 equals zero part. 𝜃 equals 90 and 𝜃 equals 270.
To solve the other part of the question, we need to calculate the inverse sine of one over root two. This is equal to 45 degrees. Therefore, one solution to this part is 45 degrees. We could draw the sine graph to calculate any other solutions. Alternatively, we could use the CAST method as shown.
There will be two solutions between zero and 360 degrees, one in the A quadrant and one in the S for sine quadrant. These will be symmetrical about the 𝑦-axis. We already know that the solution between zero and 90 is 45 degrees. This means that the solution between 90 and 180 is 135 degrees as 180 minus 45 equals 135.
There’re therefore two solutions to the equation sin 𝜃 equals one over root two between zero and 360. They are 𝜃 equals 45 degrees. And 𝜃 equals 135 degrees.
This means that the full set of values that satisfy the equation root two sin 𝜃 cos 𝜃 minus cos 𝜃 equals zero are 45 degrees, 90 degrees, 135 degrees, and 270 degrees.