Triangle PQR has a right angle Q. If sin R = 4/5, what is the value of tan P?

Explanation

The correct answer is 3/4, or 0.75. By definition of the sine ratio, since sin R = 4/5, PQ/PR = 4/5. Therefore, if PQ = 4n, then PR = 5n, where n is a positive constant. Then QR = kn, where k is another positive constant. Applying the Pythagorean theorem, the following relationship holds:(kn)2 + (4n)2 = (5n)2, or k2n2 + 16n2 = 25n2. Subtracting 16n2 from both sides of this equation yields k2n2 = 9n2. Taking the square root of both sides of k2n2 = 9n2 yields kn = 3n. It follows that k = 3. Therefore, if PQ = 4n and PR = 5n, then QR = 3n, and by definition of the tangent ratio, tan P = 3n/4n, or 3/4 or 0.75 may be entered as the correct answer.