If tan of two 𝐴 is equal to cot of 𝐴 minus 18 degrees, where two 𝐴 is an acute angle, find the value of 𝐴.
So to answer this question, we’re going to use some trigonometric identities. And your first thought may be that we need to use a double angle formula, as we have tan of two 𝐴 on the left-hand side. However, it’s actually a bit more straightforward than this, if we recall the relationship that exists between tan and cot. Remember, cot is the reciprocal of tan, is one over tan. And it’s also true that for an acute value 𝜃, tan of 𝜃 is equal to cot of 90 degrees minus 𝜃.
As two 𝐴 is an acute angle, which we’re told in the question, we can therefore substitute two 𝐴 for 𝜃 into this identity. And it gives tan of two 𝐴 is equal to cot of 90 degrees minus two 𝐴. But remember, we’re given in the question that for this value of 𝐴, tan of two 𝐴 is also equal to cot of 𝐴 minus 18 degrees. So we have cot of 90 degrees minus two 𝐴 is equal to cot of 𝐴 minus 18 degrees.
As both sides of the equation now in terms of cot, this means that the arguments of both of the trigonometric functions must be the same. So that is the bit in the brackets. We have then that 90 degrees minus two 𝐴 is equal to 𝐴 minus 18 degrees. And now, we have an equation with no trigonometric functions whatsoever. And we can solve it to find the value of 𝐴.
As we currently have terms involving 𝐴 on both sides of the equation, we can begin by adding two 𝐴 to each side. And it gives 90 degrees is equal to three 𝐴 minus 18 degrees. Next, we can add 18 degrees to each side, giving 108 degrees is equal to three 𝐴. The final step is just to divide both sides of the equation by three, which we can do using a short division. And it gives 36 degrees is equal to 𝐴. So we’ve answered the question. The value of 𝐴 is 36 degrees.