Find the value of 𝑥 given nine 𝑥 equals cos squared 60 degrees tan squared 60 degrees tan squared 30 degrees.
In this question, there are three trigonometric functions of special angles. These are results that we need to know by heart. And luckily, there’s a nice way to remember. It involves drawing a table, and we write sin 𝜃, cos 𝜃, tan 𝜃 down the left-hand side. Along the top, we add 30 degrees, 45, and 60. Then, we write one, two, three, and three, two, one. We add a denominator of two to all of these fractions. And then the final step for sin 𝜃 and cos 𝜃 is to find a square root of the numerators of each solution.
Now, of course, the square root of one is simply one. And so, sin of 30 degrees and cos of 60 degrees are both one-half. Now, tan 𝜃 is sin 𝜃 divided by cos 𝜃. So, to find tan of 30 degrees, we’ve divide sin of 30 degrees by cos of 30 degrees. But of course, since the denominators are the same, to do this, we simply divide the numerators. And so, we find tan of 30 to be one over root three, tan of 45 to be root two over root two, which is simply one, and tan of 30 to be root three over one, which is root three.
Now, we can use this table to find the values of cos of 60, tan of 60, and tan of 30 degrees. We can see that cos of 60 is equal to one-half. Tan of 60 is equal to the square root of three. And tan of 30 is equal to one over root three. Our equation says nine 𝑥 is equal to cos squared 60 — so that’s a half squared; remember, cos squared 60 is the same as cos of 60 squared — times tan squared 60, so that’s root three squared, times tan squared 30, so that’s one over root three squared.
Now, to square a fraction, we simply square the numerator and the denominator. So, a half squared is a quarter. Then, the square of a square root is simply the original number. So, the square of root three is three. Similarly, the square of one over root three is the square of one over the square of the square root of three. So, that’s one-third. And then we see that three times a third cancels to one.
So, our equation becomes nine 𝑥 equals a quarter times one or simply nine 𝑥 equals one-quarter. We solve this equation by dividing both sides by nine. But of course, dividing by nine is the same as multiplying by one-ninth. So, 𝑥 must be equal to a quarter times one-ninth, which is equal to one over 36. And so, the value of 𝑥 given that nine 𝑥 is equal to cos squared 60 times tan squared 60 times tan squared 30 is one over 36.