Video Transcript
Find the set of possible solutions of two sin 𝜃 cos 𝜃 equals zero, given 𝜃 is greater than or equal to zero and less than 360 degrees.
The square bracket tells us that 𝜃 is greater than or equal to zero degrees, whereas the curved bracket tells us that 𝜃 is strictly less than 360 degrees. We can solve our equation using our knowledge of the double-angle identities. We know that sin of two 𝜃 is equal to two sin 𝜃 cos 𝜃. This means that we need to solve sin of two 𝜃 is equal to zero. Taking the inverse sine of both sides of this equation, we get two 𝜃 is equal to the inverse sin of zero.
Whilst solving this equation would give us one solution, it is worth sketching the graphs of sin 𝜃 and sin two 𝜃 before proceeding. We know that sin 𝜃 is a periodic function, and we are interested in values between zero and 360 degrees. It has a maximum value of one and a minimum value of negative one, as shown. The graph of sin two 𝜃 will be a dilation or enlargement. The graph will be stretched by a scale factor of one-half in the horizontal direction. This means that it will look as shown.
This has a maximum value of one when 𝜃 is 45 degrees and a minimum value of negative one when 𝜃 is 135 degrees. We are interested in the points at which the graph is equal to zero. There are five such points on our graph, at zero, 90, 180, 270, and 360 degrees. We recall though that 𝜃 must be less than 360 degrees. So our last point is not a solution. The four solutions of the equation two sin 𝜃 cos 𝜃 equals zero, where 𝜃 is greater than or equal to zero degrees and less than 360 degrees, are zero, 90, 180, and 270 degrees.
