Question

When m∠A + m∠B = 90°, what relationship is formed by tan 2A and tan B? Select all that apply.

a) tan 2A = 2tan A
b) tan 2A = (2tan B) / (1 - tan^2 B)
c) tan 2A = (2tan B) / (1 + tan^2 B)
d) tan 2A = tan A + tan B

Answer 1

When angles A and B are complementary, the correct trigonometric relationship between tan 2A and tan B is tan 2A = (2tan B) / (1 - tan^2 B). Option b) is correct. Other options don't match the trigonometric identities for a double angle or complementary angles.

Step-by-step explanation:

When we know that m∠A + m∠B = 90°, angles A and B are complementary angles, and the trigonometric identities for the tangent of a double angle and the tangent of a complementary angle come into play. The tangent of a double angle is given by:

tan 2A = ​(2tan A) / (1 - tan2 A)

Since angles A and B are complementary (A + B = 90°), we have tan B = cot A, which means tan B = 1/tan A. We can use this relationship to express tan 2A in terms of tan B, leading to:

tan 2A = ​(2(1/tan B)) / (1 - (1/tan B)2)

This simplifies to:

tan 2A = ​(2tan B) / (1 - tan2 B)

Therefore, the correct relationship is option b). None of the other options accurately depict the relationship between tan 2A and tan B when A and B are complementary.

Options a), c), and d) do not correspond to the established trigonometric identities for tan 2A or tan B and should be disregarded in this context.