If cos(x)= 1/3 and x is acute, then find tan(x).

Consider a right angled triangle. Call one of the angles (not the right angle) in this triangle x. We can do this as we are told x is acute. The side opposite to x label O, the side adjacent to x label A, and label the hypotenuse H.

Now from SOHCAHTOA cos(x) = A/H = 1/3 and tan(x) = O/A . We also know by pythagoras that A+ O2 = H2 . We shall now combine these equations to get our result.

A/H = 1/3 implies H = 3A implies H2 = 9A2. Substituting this result into our euation obtained by pythagoras we get: A+ O2 = 9A2. Rearranging: O2 = 8A2 implies O2/A2 = 8 implies (O/A)2 = 8. Now we take the square root of both sides. Here we must take care, O and A are lengths and so are not negative, so we only consider the positive root: O/A = sqrt(8) = tan(x) and so we are done.

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