Which of the following angles has a positive sine and a negative secant? Is it (A) 43 degrees and 24 minutes, (B) 136 degrees and 36 minutes, (C) 223 degrees and 24 minutes, or (D) 316 degrees and 36 minutes?
Let’s begin by considering a coordinate grid. The top-right quadrant is known as quadrant one. The top-left quadrant is quadrant two. The bottom left is quadrant three. And finally, the bottom right is quadrant four. When dealing with trigonometry, these quadrants are labeled with the letters C, A, S, and T. And this is known as a CAST diagram. This helps us identify which trig functions are positive in which quadrant.
In the bottom-right quadrant, the cosine is positive, whereas the sine and tangent ratios are negative. In quadrant one, the top-right quadrant, all three of sine, cosine, and tangent are positive. In the second quadrant, the sine ratio is positive, whereas the cosine and tangent ratios are negative. Finally, in the third quadrant, the tangent ratio is positive and the sine and cosine ratios are negative. The angles in each quadrant go in a counterclockwise direction from zero to 360 degrees.
In this question, we want a positive sine ratio, which means we could be in quadrant one or quadrant two. This immediately rules out options (C) and (D) as these appear in quadrant three and four, respectively. We also want a negative secant. We know that the sec of angle 𝜃 is equal to one over the cos of angle 𝜃. As the secant of an angle and the cosine of an angle are the reciprocal of one another, we know they will either both be positive or both negative. This means that for the secant of an angle to be negative, the cosine of the angle must also be negative. This occurs in quadrant two and quadrant three.
As we need this to be true and the sine of the angle to be positive, we must be in quadrant two. This contains angles between 90 degrees and 180 degrees, so we can rule out option (A). The angle 136 degrees and 36 minutes will have a positive sine and negative secant as it lies in quadrant two.