Question

If sin theta= -5/7, which of the following are possible?

Answer 1

A) cos θ = -(√24)/7 and tan θ = 5/√24 and 
B) sec θ = 7/(√24) and tan θ = -5/(√24)

Sine is the ratio of the opposite side of an angle to the hypotenuse.  Since sin θ = -5/7, the opposite side is -5 and the hypotenuse is 7.

For A, if the cosine is -√24/7, and the sine was -5/7, then the opposite/adjacent, tangent, would be -5/-(√24).  However, two negatives divided make a positive, so the tangent would be 5/√24.

For B:  the secant is 1/cos.  This means it is the reciprocal, or flip, of the cosine.  Instead of A/H, sec = H/A.  This means the hypotenuse is 7 and the adjacent side is √24.  Tangent = opposite/adjacent; this gives us -5/√24.

For C:  the secant, H/A, is -7/5.  This means the tangent, O/A, would be -5/5 = -1.  This is incorrect.

For D:  the cosine, A/H, is √24/7.  This means the tangent, O/A, would be -5/√24; this is incorrect.