Circle the value of sin of 90 degrees. The options are zero, one, negative one, or root two over two.
90 degrees is one of those special angles for which we should know the values of its sin, cos, and tan off by heart. Other such angles are 30 degrees, 45 degrees, and 60 degrees as well as zero degrees. If you don’t know the value off by heart, then there are two methods that we can use to work it out.
The first method is to sketch a graph of 𝑦 equals sin 𝑥. We do need to know and be able to recognise what the graphs of 𝑦 equals sin 𝑥, 𝑦 equals cos 𝑥, and 𝑦 equals tan 𝑥 all look like. The maximum and minimum values of sin 𝑥 are one and negative one. So these are the values for the scale on our vertical axis.
Key points on the graph occur at multiples of 90 degrees on the horizontal axis. So on the horizontal scale, we have 90 degrees, 180 degrees, 270 degrees, 360 degrees, and so on as well as negative 90 degrees, negative 180 degrees, and so on.
Both the sin and cos graphs are wave shapes. And we need to remember that the sin graph starts at the origin. The graph then rises to one when 𝑥 is equal to 90 degrees, falls back to zero when 𝑥 is equal to 180 degrees, falls to negative one when 𝑥 is equal to 270 degrees, and rises back to zero when 𝑥 equals 360 degrees.
The graph is periodic with a period of 360 degrees, which means that this same pattern repeats for 𝑥-values greater than 360 degrees and less than zero. We want to work out the value of sin of 90 degrees. So we need to go to 90 degrees on our horizontal axis, go upwards until we meet the graph — so that’s here — and then go vertically across until we meet the 𝑦-axis. We see that the value on our vertical scale is one at this point. So the value of sine of 90 degrees taken from our graph is one.
The second method is to use a table of values and write down the value of sine for those special angles I mentioned earlier: zero, 30, 45, 60, and 90 degrees. And there’s a really nice trick that you can use to do this. Here is our table ready to go.
We begin by writing zero, one, two, three, and four horizontally across our table. We then add a denominator of two for each value. Next, we take the square root of each value in the numerator. Some of these values can be simplified. The square root of zero is just zero, the square root of one is one, and the square root of four is two. Some values can be simplified further. Zero over two is just equal to zero and two over two is equal to one.
This table now tells us the value of sine of each of these angles. For example, sin of 30 degrees is equal to one-half. Sine of 45 degrees is equal to root two over two. We were asked for the value of sine of 90 degrees. And again, we see that it’s equal to one. Learning this method will be really helpful for helping you remember the exact values of sine for each of these angles.
There’s a very similar method that you can use for remembering the values for cos. But instead of beginning with zero, one, two, three, four, we begin instead with four, three, two, one, zero or you can remember that the order of the values is just reversed.
Using both methods, we found that the value of sine of 90 degrees is one.