Which of the following is equal to tan 69 degrees? Is it A) three sin 23 degrees cos 23 degrees, B) sin 69 degrees over cos 69 degrees, C) three tan 23 degrees, or D) cos 69 degrees over sin 69 degrees?
The key to this question is knowing that tan 𝑥 is equal to sin 𝑥 over cos 𝑥. Replacing all the 𝑥s in this identity by 69 degrees, we get that tan 69 degrees is equal to sin 69 degrees over cos 69 degrees. And so our answer is B.
Let’s see where this identity comes from. We draw a right triangle and call the measure of one of the angles of that triangle 𝑥. We call the length of the hypotenuse ℎ, for hypotenuse; the length of the side adjacent to the angle we call 𝑎, for adjacent; and the length of the side opposite 𝑥 we call 𝑜, for opposite.
By definition sin 𝑥 is equal to the length of the side opposite, 𝑜, divided by the length of the hypotenuse, ℎ. Similarly, cos 𝑥 is defined to be the length of the adjacent side, 𝑎, divided by the length of the hypotenuse, ℎ. And tan 𝑥 is defined to be the length of the opposite side, 𝑜, divided by the length of the adjacent side, 𝑎.
We can use these definitions to calculate sin 𝑥 over cos 𝑥. We get 𝑜 over ℎ divided by 𝑎 over ℎ. Multiplying this by ℎ over ℎ, we get 𝑜 over 𝑎. We get 𝑜 over 𝑎, which is tan 𝑥 as required.
So that’s how you can derive the identity tan 𝑥 equals sin 𝑥 over cos 𝑥 that we used to find our answer. You can use your calculator to confirm that option B is the correct answer and that options A, C, and D are all not equal to tan 69.
In particular, the expression in option D, cos 69 degrees over 69 degrees, is one over tan 69 degrees or cot 69 degrees. And you might have got this answer had you forgotten which way round sin and cos go in our identity, and you might have thought that three times tan 23 degrees should be tan three times 23 degrees i.e., tan 69 degrees. But unfortunately, it’s not.