Find the value of one plus tan of 60 degrees tan of 30 degrees over cos squared 30 degrees.
In this question, we have three trigonometric functions of special angles. These are values we should know by heart. And luckily, there’s a really nice way to remember them. We draw a table, and on the left-hand side, we write sin 𝜃, cos 𝜃, and tan 𝜃. And along the top, we have 30 degrees, 45, and 60. The first two rows are really straightforward. We write one, two, three, then three, two, one. We then make these a fraction with a denominator of two. And then the final step is to find the square root of all of the numerators. But of course, the square root of one is simply one.
And so, there’s really no need to add this. And we have the values for sin 𝜃 and cos 𝜃 when 𝜃 is 30, 45, and 60. Tan 𝜃 is not quite a straightforward but to find it, we divide the values for sin 𝜃 by the values for cos 𝜃. So, for example, to find tan of 30, we divide one-half by root three over two. And since the denominators are equal, we can simply divide the numerators. So, tan of 30 is one over root three. Then, tan of 45 is root two over root two, which is simply one. And tan of 60 is root three over one, which is the square root of three.
And so, we have the values for sin, cos, and tan of 30, 45, and 60. We can find tan of 60 in our table, and it’s the square root of three. The next value we’re interested in in the question is tan of 30. So, that’s one over the square root of three. And to find cos squared of 30, we simply find cos of 30 and then square it. Well, cos of 30 is the square root of three over two, so cos squared of 30 or cos 30 all squared is the square root of three over two all squared, which is three over four. Remember, this is simply because the square root of three squared is three, and two squared is four.
We substitute each of these back into our original expression. So, we get one plus the square root of three times one over the square root of three all over three-quarters. Well, the square root of three times one over the square root of three is simply one. So, we get two over three-quarters. But of course, to divide by a fraction, we multiply by the reciprocal of that fraction. So, we’re going to do two multiplied by four-thirds. It might help to write two as two over one, and then we simply multiply the numerators and multiply the denominators. And when we do, we find the value of one plus tan of 60 times tan of 30 over cos squared 30 to be eight-thirds.