Determine whether the following statement is true or false: Sin of 30 degrees is greater than sin of 60 degrees.
Now, presumably, we don’t have access to a calculator in this question. Because it would be far too straightforward to simply type sin of 30 degrees and sin of 60 degrees into our calculator and work out which is bigger. Instead, we’ll consider two methods that we can use to answer this question when we don’t have access to a calculator.
The first method is a graphical method. We’ll sketch the graph of 𝑦 equals sin 𝑥 and use this to help us determine whether the statement is true or false. We do need to know what the graph of 𝑦 equals sin 𝑥 looks like and be able to sketch it from memory. The maximum and minimum values of the graph of 𝑦 equals sin 𝑥 are one and negative one. Key points on the 𝑥-axis of this graph are multiples of 90 degrees, 90 degrees, 180 degrees, 270 degrees, 360 degrees, and so on.
The graph of 𝑦 equals sin 𝑥 begins at the origin. It’s, then, a wave shape rising to one at 90 degrees, falling back to zero at 180 degrees, falling to negative one at 270 degrees, and rising back to zero again at 360 degrees. The graph is periodic with a period of 360 degrees. So this same shape then repeats to the right of 360 degrees and to the left of zero. Now, let’s see how we could use this graph to help us determine whether the statement is true or false.
Between zero and 90 degrees, the graph of 𝑦 equals sin 𝑥 is increasing. The values are getting larger. 30 degrees and 60 degrees on the 𝑥-axis would be about here and here. Looking across to the 𝑦-axis, we can see that the value for sin of 60 degrees is larger than the value for sin of 30 degrees. From our graph, we see that sin of 30 degrees is, in fact, less than sin of 60 degrees. And therefore, the original statement, which said that sin of 30 degrees was greater than sin of 60 degrees, is false.
The other method we can use is to work out the values of sin of 30 degrees and sin of 60 degrees. But, remember I said we wouldn’t have access to a calculator. However, we should know the values of sin of 30 degrees and sin of 60 degrees. Because they’re special angles for which their sines, cosines, and tans can be expressed exactly in terms of fractions and surds.
There’s a helpful method we can use for remembering these values. And we’ll also include an angle of 45 degrees. We first write one, two, and three. We then divide each of these values by two and then take the square root of each of the numerators, giving root one over two, root two over two, and root three over two. The square root of one is just one. So, our value for sin of 30 degrees simplifies to one over two. But the square root of two and the square root of three can’t be simplified. So, we have that the sin of 30 degrees is equal to one-half and sin of 60 degrees is equal to root three over two. And now we need to compare these values.
The question is, is one-half greater than root three over two? As the denominators of these two fractions are the same, we can actually just compare the numerators. And so now we’re asking, is one greater than root three? Now, you may know the square root of three approximately. It’s approximately equal to 1.73, which you may be familiar with if you’ve done a lot of work with exact trigonometric values. But if you don’t, we can work this out in a different way.
Both one and the square root of three are positive. So, both sides of this inequality are positive, which means we can square both sides. And it won’t affect the inequality. One squared is one, and the square root of three squared is three. So, the question has now become, is one greater than three? Well, clearly, no. One is not greater than three, which means the original question we asked, which was “was one-half greater than root three over two”, was false. Therefore, we conclude again that sin of 30 degrees is less than sin of 60 degrees. And therefore, the statement we were given is false.