Determine the quadrant in which 𝜃 lies if the cos of 𝜃 is greater than zero and the sin of 𝜃 is greater than zero.
We can begin here by sketching a coordinate grid and labeling the four quadrants. Quadrant one is the top right; quadrant two, the top left; quadrant three, the bottom left; and quadrant four, the bottom right. When dealing with trigonometry, these quadrants are labeled C, A, S, and T. And this is known as a CAST diagram. This enables us to quickly see which of our trig ratios are positive or negative.
In the fourth quadrant labeled C, the cosine of any angle is positive, whereas the sine and tangent of any angle is negative. In the first quadrant, all three ratios are positive. This means that the sine, cosine, or tangent of any angle between zero and 90 degrees is positive. The S in quadrant two tells us that the sine of any angle between 90 and 180 degrees is positive, whereas the cosine or tangent of an angle here is negative. Finally, in quadrant three, the tangent is positive, whereas the sine and cosine are negative.
In this question, we are told that cos of 𝜃 is greater than zero and sin of 𝜃 is greater than zero. This means that we are looking for the quadrant where sin 𝜃 and cos of 𝜃 are both positive. This only occurs when 𝜃 is between zero and 90 degrees. Therefore, the correct answer is the first quadrant.