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.
