If sin theta= -5/7, which of the following are possible? a. cos theta =-√24/7, and tan theta=5/√24 b sec theta = 7/√24, and tan theta= -5/√24 c. sec theta = -7/5, and tan theta = 5/√24 d cos theta = √24/7, and tan theta= 5/√24

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.

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