Which of these values cannot be the sine of an angle? Circle your answer. The options are negative 0.88, 1.05, negative one, or 0.22.
To answer this question, we can picture what the graph of 𝑦 equals sin 𝑥 looks like. It’s a smooth curve, starting at the origin. It then rises to one when 𝑥 is equal to 90 degrees. The graph then falls back to zero, reaching the 𝑥-axis when 𝑥 is 180 degrees. The graph then falls to negative one, reaching this value when 𝑥 is equal to 270 degrees, and it then rises back to zero again, reaching the 𝑥-axis when 𝑥 is 360 degrees. The graph is periodic with a period of 360 degrees, meaning the same section of the graph repeats every 360 degrees.
Notice that the graph is completely contained between the values of negative one and one vertically. That’s called the range of this function. Sin of any angle 𝑥 is always between the values of negative one and one, inclusive. It isn’t possible to have sin of 𝑥 equalling two or negative 3.8, for example.
Looking at the four values we’ve been given, we can see that only one of these four values is outside the range of negative one to one, the value of 1.05. So this value cannot be the sine of an angle.
There is another way to see that the sine of an angle can never be more than one, if we consider a right-angled triangle, which is probably how you first learnt about sine. The definition of sin of an angle 𝑥 degrees is as the opposite side of a right-angled triangle divided by its hypotenuse.
But remember, the hypotenuse of a right-angled triangle is always its longest side, which means it’s longer than the opposite. So we’re dividing something by something larger than itself, which will always give an answer less than one.
The possibilities of sine being equal to one or between negative one and zero inclusive arise when we extend our definition of sine into the unit circle and to angles which are greater than 90 degrees or less than zero. By using the graph of 𝑦 equals sin 𝑥 or by picturing a right-angled triangle, we can see that the value 1.05 cannot be the sine of any angle.