Find the set of values satisfying the equation five cos squared 𝜃 equals four, where zero degrees is less than or equal to 𝜃 which is less than 360 degrees. Give the answer to the nearest minute.
We’re being asked to solve the trigonometric equation five cos squared 𝜃 equals four for 𝜃, where 𝜃 is greater than or equal to zero and less than 360 degrees. So, where do we start? Well, we begin by solving just like any other equation. We’re going to divide both sides by five such that cos squared 𝜃 is four-fifths or 0.8. Next, we take the square root of both sides of our equation, remembering to take both the positive and negative square root of four-fifths. The square root of four-fifths can be written as two root five over five. So, we have two equations that we need to solve, cos of 𝜃 equals two root five over five or negative two root five over five.
We solve for 𝜃 by taking the inverse cos of both sides of our equation. When we take the inverse cos of two root five over five, making sure our calculator is in degree mode, we get 26.5650 and so on. And the inverse cos of negative two root five over five is 153.4349 and so on. Now, we’re going to leave those values unrounded for now, but we’re not quite finished. We wanted to find the set of values for 𝜃 is greater than or equal to zero and less than 360 degrees. And so, there are a couple of ways we can find the other solutions.
One way is to think about the shape of the graph 𝑦 equals cos of 𝑥. In the interval from 𝜃 is greater than or equal to zero to less than 360, it looks a little like this with maximums at one and minimums at negative one. The line 𝑦 equals two root five over five looks a little something like this. We know it has one solution at 26.5 and so on degrees. Now, we can see that the graph has reflection or symmetry about the line 𝑥 equals 180 degrees, so we find the other value of 𝜃 by subtracting 26.5 and so on from 360 degrees. That gives us 333.4349 and so on.
Similarly, if we sketch the line, 𝑦 equals negative two root five over five, it looks a bit like this. This time, one solution lies at 153.4 degrees and so one. To find the fourth solution, we subtract this from 360. That gives us 206.5650 and so on. Now, we want to round our answers to the nearest minute. We could multiply the decimal part by 60 to achieve this. Alternatively, there’s a button on most calculators that will do this for us. And it looks a little something like this. When we press this button for our first value, we get 26 degrees and 34 minutes correct to the nearest minute. We get 153 degrees and 26 minutes for our next solution. And our other two solutions are 260 degrees and 34 minutes and 333 degrees and 26 minutes. And so, the set of values that satisfy our solution are shown.
Now, there was another way we could have found these values, and that was to use the CAST diagram. The CAST diagram looks like this. We put the letters C-A-S-T, and this shows us where our values for cos 𝜃, sin 𝜃, tan 𝜃, or all three are positive. Now, our first solution to cos 𝜃 equals two root five over five was 26.56 and so on degrees. The other solution will be here in the fourth quadrant where cos 𝜃 is positive. We find this value of 𝜃 by subtracting 26.56 from 360. And that gives us, to the nearest minute, 333 degrees and 26 minutes. Our other solution was 𝜃 equals 153.43 and so on. Cos 𝜃 is also negative in this third quadrant, so we subtract 153.43 and so on from 360 degrees. And once again, we get 206 degrees and 34 minutes correct to the nearest minute.