Find the value of 𝜃 given two cos 𝜃 plus one equals zero, where 𝜃 is the largest angle in the range from zero degrees to 360 degrees inclusive.
Our first step is to solve the equation two cos 𝜃 plus one equals zero. Subtracting one from both sides of the equation gives us two cos 𝜃 equals negative one. And dividing both sides of this equation by two gives us cos 𝜃 is equal to negative a half. One value of 𝜃 will therefore be given by cos minus one or inverse cos of negative a half. Typing this into the calculator gives us a value of 𝜃 of 120 degrees.
This question asked us to find the largest angle in the range from zero degrees to 360 degrees. If we consider the graph of cos 𝜃, we can see that it is equal to negative a half at two points. The first point is the value we’ve already worked out — 120 degrees. There is also a second point between 180 degrees and 270 degrees.
As the cosine graph is symmetrical about 180 degrees, we can work out the second answer by subtracting 120 degrees from 360 degrees. This gives us 𝜃 equals 240 degrees.
There are two solutions to the equation two cos 𝜃 plus one equals zero in the range zero to 360 degrees. These are 120 degrees and 240 degrees, of which the largest angle is 240 degrees.