Question Video: Identifying the Quadrant Where an Angle Lies given the Signs of Its Trigonometric Functions Mathematics

The angle 𝜃 is in standard position where the sec 𝜃 < 0. In which quadrants does the terminal side of 𝜃 lie?

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Video Transcript

The angle 𝜃 is in standard position, where the sec of 𝜃 is less than zero. In which quadrants does the terminal side of 𝜃 lie?

We know that an angle is in standard position in the coordinate plane if its vertex is located at the origin and one ray is on the positive 𝑥-axis. The ray on the 𝑥-axis is called the initial side and the other ray the terminal side. The initial side is where the angle starts, and the terminal side is the ray where the measurement of the angle stops.

We can begin by sketching a coordinate grid and labeling the four quadrants as shown. Quadrant I is the top right, quadrant II the top left, quadrant III is the bottom left, and quadrant IV is the bottom right. When the angle 𝜃 is in standard position, our angles go from zero to 360 degrees in a counterclockwise direction as shown.

When dealing with trigonometry, the quadrants are labeled with the letters C, A, S, and T, and this is known as a CAST diagram. This enables us to quickly identify the quadrants where the sine, cosine, and tangent ratios are positive and negative.

In quadrant IV, the C tells us that the cosine of any angle in this quadrant will be positive, whereas the sine and tangent of an angle will be negative. In the first quadrant, all of the ratios will be positive. This means that the sine, cosine, and tangent of any angle between zero and 90 degrees will be positive. In the second quadrant, the sine of an angle will be positive, whereas the cosine and tangent will be negative. Finally, in the third quadrant, the tangent is positive, whereas the sine and cosine are negative.

In this question, we are told that the sec of angle 𝜃 is less than zero. This means it is negative. We know that the sec of angle 𝜃 is equal to one over the cos of angle 𝜃. This means that they are the reciprocal of one another. If the cos of 𝜃 is positive, the sec of 𝜃 will be positive. And if the cosine is negative, the secant will be negative.

In this question, we are looking for the quadrants where the cos of 𝜃 is negative. This is true in quadrant II and quadrant III, whereas the cos of 𝜃 is positive in quadrant I and IV. The correct answer, therefore, is second or third. The terminal side of angle 𝜃 could lie in the second or third quadrants.

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